[1] Abel, N. H., Recherches sur les fonctions elliptiques, J. Reine Angew. Math. 2 (1827), 101–181; 3 (1828), 160–190; Oeuvres 1, pp. 263–388.
[2] Abel, N. H., Sur une espèce de fonctions entières nées du developpement de la fonction (1-v)-1e- xv/1-v suivant les puissances de v, Oeuvres 2 (1881), p. 284.
[3] Ablowitz, M. J. and Segur, H., Exact linearization of a Painlevé transcendent, Phys. Rev. Lett. 38 (1977), 1103–1106.
[4] Abramowitz, M. and Stegun, I., Handbook of Mathematical Functions, Dover, Mineola, NY 1965.
[5] Airault, H.Rational solutions of Painlevé equations, Stud. Appl. Math. 61 (1979), 31–53; (correction) 64 (1981), 183.
[6] Airy, G. B., On the intensity of light in the neighborhood of a caustic, Trans. Camb. Philos. Soc. VI (1838), 379–402.
[7] Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, NY 1965.
[8] Akhiezer, N. I., Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI 1990.
[9] Al-Salam, W. A., Characterization theorems for orthogonal polynomials, Orthogonal Polynomials, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 294, Kluwer, Dordrecht 1990, pp. 1–24.
[10] Andrews, G. E., Askey, R., and Roy, R., Special Functions, Cambridge University Press, Cambridge 1999.
[11] Andrews, L. C., Special Functions of Mathematics for Engineers, 2nd ed., McGraw-Hill, New York, NY 1992.
[12] Aomoto, K., Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987), 545–549.
[13] Appell, P.Sur les fonctions hypergéométriques de deux variables. J. Math. Pures Appl. 8 (1882), 173–216.
[14] Appell, P. and Kampé de Fériet, J., Fonctions Hypergéométriques de Plusieurs Variables: Polynˆomes d'Hermite, Gauthier-Villars, Paris 1926.
[15] Appell, P. and Lacour, E., Principes de la Théorie des Fonctions Elliptiques et Applications, Gauthier-Villars, Paris 1922.
[16] Armitage, J. V. and Eberlein, W. F., Elliptic Functions, Cambridge University Press, Cambridge 2006.
[17] Artin, E., The Gamma Function, Holt, Rinehart and Winston, New York, NY 1964.
[18] Askey, R., Orthogonal Polynomials and Special Functions, Society for Industrial and Applied Mathematics, Philadelphia, PA 1975.
[19] Askey, R., Continuous Hahn polynomials, J. Phys. A 18 (1985), L1017–1019.
[20] Askey, R., Handbooks of special functions, A Century of Mathematics in America, vol. 3, P., Duren, ed., American Mathematical Society, Providence, RI 1989, pp. 369–391.
[21] Askey, R. and Wilson, J. A., A set of orthogonal polynomials that generalize the Racah coefficients or 6j symbols, SIAM J. Math. Anal. 10 (1979), 1008–1016.
[22] Askey, R. and Wilson, J. A., Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials, Memoirs of the AmericanMathematical Society, 54, American Mathematical Society, Providence, RI 1985.
[23] Atakishiyev, N. M., Rahman, M., and Suslov, S. K., On classical orthogonal polynomials, Constr. Approx. 11 (1995), 181–226.
[24] Atakishiyev, N. M. and Suslov, S. K., The Hahn and Meixner polynomials of an imaginary argument and some of their applications, J. Phys. A 18 (1985), 1583–1596.
[25] Ayoub, R., The lemniscate and Fagnano's contribution to elliptic integrals, Arch. Hist. Exact Sci. 29 (1983/84), 131–149.
[26] Bäcklund, A. V., Om ytor med konstant negativ krokning, Lunds Univ. Arsskr. 19 (1883), 1–41.
[27] Baik, J., Kriecherbauer, T., McLaughlin, K., and Miller, P. D., Discrete Orthogonal Polynomials: Asymptotics and Applications, Princeton University Press, Princeton, NJ 2007.
[28] Bailey, W. N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge 1935.
[29] Baker, H. F., Abelian Functions: Abel's Theory and the Allied Theory of Theta Functions, Cambridge University Press, New York, NY 1995.
[30] Barnes, E. W., A new development of the theory of the hypergeometric functions, Proc. London Math. Soc. 50 (1908), 141–177.
[31] Bassett, A. B., On the potentials of the surfaces formed by the revolution of limaҫons and cardioids, Proc. Camb. Philos. Soc. 6 (1889), 2–19.
[32] Bassom, A. P., Clarkson, P. A., Law, C. K., and McLeod, J. B., Application of uniform asymptotics to the second Painlevé transcendent, Arch. Rational Mech. Anal. 143 (1998), 241–271.
[33] Beals, R., Gaveau, B., and Greiner, P. C., Uniform hypoelliptic Green's functions, J. Math. Pures Appl. 77 (1998), 209–248.
[34] Beals, R. and Kannai, Y., Exact solutions and branching of singularities for some equations in two variables, J. Diff. Equations 246 (2009), 3448–3470.
[35] Beals, R., and Sattinger, D. H., Integrable systems and isomonodromy deformations, Physica D 65 (1993), 17–47.
[36] Beals, R., and Szmigielski, J., Meijer G-functions: a gentle introduction, Notices Amer. Math. Soc. 60 (2013), 866–872.
[37] Bernoulli, D., Solutio problematis Riccatiani propositi in Act. Lips. Suppl. Tom VIII p. 73, Acta Erud. Publ. Lipsiae (1725), 473–475.
[38] Bernoulli, D., Demonstrationes theorematum suorum de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae, Comm. Acad. Sci. Petr. 7 (1734–35, publ. 1740), 162–173.
[39] Bernoulli, Jac., Curvatura laminae elasticae, Acta Erud. (1694), 276–280; Opera I, pp. 576–600.
[40] Bernoulli, Jac., Ars Conjectandi, Basel 1713.
[41] Bernoulli, Joh., Methodus generalis construendi omnes aequationes differentialis primi gradus, Acta Erud. Publ. Lipsiae (1694), 435–437.
[42] Bernstein, S., Sur les polynomes orthogonaux relatifs à un segment fini, J. de Math. 9 (1930), 127–177; 10 (1931), 219–286.
[43] Bessel, F. W., Untersuchung des Theils der planetarischen Störungen aus der Bewegung der Sonne entsteht, Berliner Abh. (1824), 1–52.
[44] Beukers, F., Hypergeometric functions, how special are they? Notices Amer. Math. Soc. 61 (2014), 48–56
[45] Binet, J. P. M., Mémoire sur les intégrales définies eulériennes et sur leur application à la théorie des suites ainsi qu’à l’évaluation des fonctions des grandes nombres, J. École Roy. Polyt. 16 (1838–9), 123–143.
[46] Birkhoff, G. D., Formal theory of irregular linear difference equations, Acta Math. 54 (1930), 205–246.
[47] Birkhoff, G. D. and Trjitzinsky, W. J., Analytic theory of singular difference equations, Acta Math. 60 (1933), 1–89.
[48] Bleistein, N. and Handelsman, R. A., Asymptotic Expansions of Integrals, Dover, Mineola, NY 1986.
[49] Bobenko, A. I. and Eitner, U., Painlevé Equations in the Differential Geometry of Surfaces, Springer, Berlin 2000.
[50] Bochner, S., Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730–736.
[51] Bohr, H. and Mollerup, J., Laerebog i Matematisk Analyse, vol. 3, J. Gjellerup, Copenhagen 1922.
[52] Boutroux, P., Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du seconde ordre, Ann. Sci. École Norm. Sup. 30 (1913), 255–375; 31 (1914), 99–159.
[53] Brillouin, L., Remarques sur la mécanique ondulatoire, J. Phys. Radium 7 (1926), 353–368.
[54] Briot, C. and J.-C., Bouquet, Mémoire sur l'intégration des équations différentielles au moyen des fonctions elliptiques, J. École Imp. Poly. 36 (1856), 199–254.
[55] Bruno, A. D. and Batkhin, A. B., Painlevé Equations and Related Topics, de Gruyter, Berlin 2012.
[56] Brychkov, Yu. A., Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, Boca Raton, FL 2008.
[57] Buchholz, H., The Confluent Hypergeometric Function with Special Emphasis on its Applications, Springer, New York, NY 1969.
[58] Burchnall, J. L., Differential equations associated with hypergeometric functions. Quart. J. Math. Oxford 13 (1942), 90–106.
[59] Burchnall, J. L. and Chaundy, T. W., Expansions of Appell's double hypergeometric functions, II, Quart. J. Math. Oxford 12 (1941), 112–128.
[60] Burkhardt, H., Entwicklungen nach oscillierenden Funktionen und Integration der Differentialgleichungen der mathematischen Physik, Deutsche Math. Ver. 10 (1901–1908), 1–1804.
[61] Campbell, R., Les Intégrales Euleriennes et leurs Applications, Dunod, Paris 1966.
[62] Cao, L.-H., Li, Y.-T., and Lin, Y., Asymptotic approximations of the continuous Hahn polynomials and their zeros, J. Approx. Theory, to appear.
[63] Carleman, T., Sur le problème des moments, C. R. Acad. Sci. Paris 174 (1922), 1680.
[64] Carlini, F., Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero, Appendice all’ Effemeridi Astronomiche di Milano per l'Anno 1818, Milan 1817, pp. 3–48.
[65] Carlson, B. C., Special Functions of Applied Mathematics, Academic Press, New York, NY 1977.
[66] Chandrasekharan, K., Elliptic Functions, Springer, Berlin 1985.
[67] Charlier, C. V. L., Über die Darstellung willkürlichen Funktionen, Ark. Mat. Astr. Fysic 2 (1905–6), 1–35.
[68] Chaundy, T. W.An extension of hypergeometric functions, I, Quart. J. Math. Oxford 14 (1943), 55–78.
[69] Chebyshev, P. L., Théorie des mécanismes connus sous le nom de parall élogrammes, Publ. Soc. Math. Warsovie 7 (1854), 539–568; Oeuvres 1, pp. 109–143.
[70] Chebyshev, P. L., Sur les fractions continues, Utzh. Zap. Imp. Akad. Nauk 3 (1855), 636–664; J. Math. Pures Appl. 3 (1858), 289–323; Oeuvres 1, pp. 201–230.
[71] Chebyshev, P. L., Sur une nouvelle série, Bull. Phys. Math. Acad. Imp. Sci. St. Pét. 17 (1858), 257–261; Oeuvres 1, pp. 381–384.
[72] Chebyshev, P. L., Sur le développment des fonctions à une seule variable, Bull. Phys. Math. Acad. Imp. Sci. St. Pét. 1 (1859), 193–200; Oeuvres 1, pp. 499–508.
[73] Chebyshev, P. L., Sur l'interpolation des valeurs équidistantes, Zap. Akad. Nauk 4 (1864); Oeuvres 1, pp. 219–242.
[74] Chebyshev, P. L., Sur les valeurs limites des intégrales, J. Math. Pures Appl. 19 (1874), 193–200.
[75] Chester, C., Friedman, B., and Ursell, F., An extension of the method of steepest descents, Proc. Camb. Philos. Soc. 53 (1957), 599–611.
[76] Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, NY 1978.
[77] Christoffel, E. B., Ueber die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 61–82.
[78] Christoffel, E. B., Ueber die lineare Abhängigkeit von Funktionen einer einzigen Veränderlichen, J. Reine Angew. Math. 55 (1858), 281–299.
[79] Christoffel, E. B., Sur une classe particulière de fonctions entières et de fractions continues, Ann. Mat. Pura Appl. 8 (1877), 1–10.
[80] Chu, S.-C., Ssu Yuan Yü Chien (Precious Mirror of the Four Elements), 1303.
[81] Clancey, K. F. and Gohberg, I, Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser, Basel, 1981.
[82] Clarkson, P. A., Painlevé equations – nonlinear special functions, Orthogonal Polynomials and Special Functions: Computation and Applications, F., Marcellán and W., Van Assche, eds., Springer, Berlin 2006, pp. 331–411.
[83] Clausen, T., Ueber die Fälle, wenn die Reihe von der Form. ein Quadrat von der Form. hat, J. Reine Angew. Mat. 3 (1828), 93–96.
[84] Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY 1955.
[85] Conte, R., and Musette, M., The Painlevé Handbook, Springer, Dordrecht 2008.
[86] Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, Oxford 1955.
[87] Copson, E. T., On the Riemann–Green function, Arch. Rat. Mech. Anal. 1 (1958), 324–348.
[88] Copson, E. T., Asymptotic Expansions, Cambridge University Press, Cambridge 2004.
[89] Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. 1, Wiley, New York, NY 1961.
[90] Cuyt, A., Petersen, V. B., Verdonk, B., Waadeland, H., and Jones, W. B., Handbook of Continued Fractions for Special Functions, Springer, New York, NY 2008.
[91] Dai, D., Ismail, M. E. H., and Wang, X.-S., Plancherel–Rotach asymptotic expansion for some polynomials from indeterminate moment problems, Constr. Approx. 40 (2014), 61–104.
[92] D'Antonio, L., Euler and elliptic integrals, Euler at 300: An Appreciation, Mathematical Association of America, Washington, DC 2007, pp. 119–129.
[93] Darboux, G., Mémoire sur l'approximation des fonctions de très-grandes nombres et sur une classe étendue de développements en série, J. Math. Pures Appl. 4 (1878), 5–57.
[94] Darboux, G., Théorie Générale des Surfaces, vol. 2, book 4, Gauthier-Villars, Paris 1889.
[95] Davis, P. J., Leonhard Euler's integral: a historical profile of the gamma function. In memoriam: Milton Abramowitz, Amer. Math. Monthly 66 (1959), 849–869.
[96] de Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland, Amsterdam 1961.
[97] Debye, P., Näherungsformeln für die Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche Werte des Index, Math. Ann. 67 (1909), 535–558.
[98] Deift, P., Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach, American Mathematical Society, Providence, RI 1999.
[99] Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S., and Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 70 (1999), 1491–1552.
[100] Deift, P., and Zhou, X., A steepest descentmethod for oscillatory Riemann–Hilbert problems: Asymptotics for the MKdV equation, Ann. Math. 137 (1993), 295–368.
[101] Deift, P., and Zhou, X., Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), 277–377.
[102] Delache, S., and Leray, J., Calcul de la solution élémentaire de l'opérateur d'Euler–Poisson–Darboux et de l'opérateur de Tricomi–Clairaut hyperbolique d'ordre 2, Bull. Soc. Math. France 99 (1971), 313–336.
[103] Dieudonné, J., Abrégé d'Histoire des Mathématiques, vols. 1 and 2, Hermann, Paris 1978.
[104] Dieudonné, J., Special Functions and Linear Representations of Lie Groups, CBMS Series, 42, American Mathematical Society, Providence, RI 1980.
[105] Dirichlet, G. L., Sur les séries dont le terme général dépend de deux angles, et qui servent à exprimer des fonctions arbitraires entre les limites données, J. für Math. 17 (1837), 35–56.
[106] Dubrovin, B., Theta functions and nonlinear equations, Russian Math. Surv. 36 (1981), 11–80.
[107] Dunkl, C., A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J., 25 (1976), 335–358.
[108] Dunster, T. M., Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory 112 (2001), 93–133.
[109] Dutka, J., The early history of the hypergeometric function, Arch. Hist. Exact Sci. 31 (1984/85), 15–34.
[110] Dutka, J., The early history of the factorial function, Arch. Hist. Exact Sci. 43 (1991/92), 225–249.
[111] Dutka, J., The early history of Bessel functions, Arch. Hist. Exact Sci. 49 (1995/96), 105–134.
[112] Dwork, B., Generalized Hypergeometric Functions, Oxford University Press, New York, NY 1990.
[113] Edwards, H. M., Riemann's Zeta Function, Dover, Mineola, NY 2001.
[114] Erdélyi, A., Über eine Integraldarstellung der Mk,m-Funktion und ihre aymptotische Darstellung für grosse Werte von R(k), Math. Ann. 113 (1937), 357–361.
[115] Erdélyi, A., Asymptotic Expansions, Dover, Mineola, NY 1956.
[116] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher Transcendental Functions, vols. I–III, Robert E. Krieger, Melbourne, FL 1981.
[117] Erdélyi, A., Magnus, W., and Oberhettinger, F., Tables of Integral Transforms, vols. I and II, McGraw-Hill, New York, NY 1954.
[118] Erdõs, P. and Turán, P., On interpolation. III: Interpolatory theory of polynomials, Ann. Math. 41 (1940), 510–553.
[119] Euler, L., De progressionibus transcendentibus seu quarum termini generales algebraice dari nequent, Comm. Acad. Sci. Petr. 5 (1738), 36–57; Opera Omnia I vol. 14, pp. 1–24.
[120] Euler, L., De productis ex infinitus factoribus ortis, Comm. Acad. Sci. Petr. 11 (1739), 3–31; Opera Omnia I vol. 14, pp. 260–290.
[121] Euler, L., De motu vibratorum tympanorum, Comm. Acad. Sci. Petr. 10 (1759), 243–260; Opera Omnia II, vol. 10, pp. 344–358.
[122] Euler, L., De integratione aequationis differentialis mdx, Nov. Comm. Acad. Sci. Petr. 6 (1761), 37–57; Opera Omnia I vol. 20, pp. 58–79.
[123] Euler, L., Institutiones Calculi Integralis, vol. 2, St. Petersburg 1769; Opera Omnia I vol. 12, pp. 221–230.
[124] Euler, L., Evolutio formulae integralis integratione a valore x=0 ad x = 1 extensa, Nov. Comm. Acad. Sci. Petr. 16 (1771), 91–139; Opera Omnia I, vol. 17, pp. 316–357.
[125] Euler, L., Demonstratio theorematis insignis per conjecturam eruti circa integrationem formulae, Institutiones Calculi Integralis, vol. 4, St. Petersburg 1794, pp. 242–259; Opera Omnia I, vol. 19, pp. 197–216.
[126] Euler, L., Methodus succincta summas serierum infintarum per formulas differentiales investigandi, Mém. Acad. Sci. St. Pét. 5 (1815), 45–56; Opera Omnia I, vol. 16, pp. 200–213.
[127] Fabry, E., Sur les intégrales des équations différentielles à coefficients rationnels, Dissertation, Paris 1885.
[128] Fagnano dei Toschi, G. C., Produzioni Matematiche, Gavelliana, Pesaro 1750.
[129] Favard, J., Sur les polynˆomes de Tchebicheff, C. R. Acad. Sci. Paris 200 (1935), 2052–2053.
[130] Ferreira, C., López, J. L., and Pagola, P. J., Asymptotic approximations between the Hahn-type polynomials and Hermite, Laguerre and Charlier polynomials, Acta Appl. Math. 103 (2008), 235–252.
[131] Ferreira, C., López, J. L., and Sinusía, E. P., Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials, J. Comput. Appl. Math. 217 (2008), 88–109.
[132] Fejér, L., Sur une méthode de M. Darboux, C. R. Acad. Sci. Paris 147 (1908), 1040–1042.
[133] Fejér, L., Asymptotikus értékek meghatározáráról (On the determination of asymptotic values), Math. Termész. Ért. 27 (1909), 1–33; Ges. Abh. I, pp. 445–503.
[134] Ferrers, N. M., An Elementary Treatise on Spherical Harmonics and Functions Connected with Them, MacMillan, London 1877.
[135] Fine, N. J., Basic Hypergeometric Series and Applications, American Mathematical Society, Providence, RI 1988.
[136] Flaschka, H., and Newell, A. C., Monodromy and spectrum-preserving deformations I, Comm. Math. Phys. 61 (1980), 65–116.
[137] Fokas, A. S., Its, A. R., Kapaev, A. A., and Novokshenov, V. Y., Painlevé Transcendents: the Riemann–Hilbert Approach, American Mathematical Society, Providence, RI 2006.
[138] Fokas, A. S., Its, A. R., and Kataev, A. V., An isomondromy approach to the problem of two-dimensional quantum gravity, Uspekhi Mat. Nauk 45 (1990), 135–136; Russian Math. Surv. 45 (1990), 155–157.
[139] Forrester, P. J. and Warnaar, S. O., The importance of the Selberg integral, Bull. Amer. Math. Soc. 45 (2008), 489–534.
[140] Forsyth, A. R., A Treatise on Differential Equations, Dover, Mineola, NY 1996.
[141] Fourier, J. B. J., La Théorie Analytique de la Chaleur, Firmin Didot, Paris 1822.
[142] Fox, C., The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1963), 395–429.
[143] Fox, C., Integral transforms based upon fractional integration, Proc. Camb. Philos. Soc. 59 (1963), 63–71.
[144] Freud, G., Orthogonal Polynomials, Pergamon, New York, NY 1971.
[145] Fuchs, R., Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei in Endlichen gelegenden wesentlichen Singularitäten, Math. Ann. 63 (1907), 301–321.
[146] Gambier, B., Sur les équations du seconde ordre et du premier degré dont l'intégrale générale est à points critiques fixes, Acta Math. 33 (1910), 1–55.
[147] Gardner, C., Greene, J., Kruskal, M., and Miura, R., Korteweg–deVries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math. 27 (1974), 97–133.
[148] Garnier, R., Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d’équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes, Ann. Sci. École Norm. Sup. 29 (1912), 1–126.
[149] Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge 2004.
[150] Gauss, C. F., Disquisitiones generales circa seriem infinitam, Comm. Soc. Reg. Sci. Gött. 2 (1813), 46pp.; Werke 3, pp. 123–162.
[151] Gauss, C. F., Methodus nova integralium valores per approximationem inveniendi, Comm. Soc. Reg. Sci. Gött. 3 (1816), 39–76; Werke 3, pp. 163–196.
[152] Gauss, C. F., Determinatio attractionis quam in punctum quodvis positionis datae exerceret platea si ejus massa per totam orbitam ratione temporis quo singulae partes discribuntur uniformiter esset dispertita, Comm. Soc. Reg. Sci. Gött. 4 (1818), 21–48; Werke 3, pp. 332–357.
[153] Gauss, C. F., Lemniscatische Funktionen I, Werke 3, 1876, pp. 404–412.
[154] Gauss, C. F., Kugelfunktionen, Werke 5, 1877, pp. 630–632.
[155] Gautschi, W., Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys., 38 (1959), 77–81.
[156] Gautschi, W., Orthogonal Polynomials: Computation and Approximation, Oxford University Press, New York, NY 2004.
[157] Gautschi, W., Leonhard Euler: his life, the man, and his works, SIAM Review 50 (2008), 3–33.
[158] Gegenbauer, L., Über die Bessel'schen Functionen, Wiener Sitzungsber. 70 (1874), 6–16.
[159] Gegenbauer, L., Über einige bestimmte Itegrale, Wiener Sitzungsber. 70 (1874), 434–443.
[160] Gegenbauer, L., Über die Funktionen Cnv (x), Wiener Sitzungsber. 75 (1877), 891–896; 97 (1888), 259–316.
[161] Gegenbauer, L., Das Additionstheorem der Funktionen Cnv, Wiener Sitzungsber. 102 (1893), 942–950.
[162] Gelfand, I. M., Graev, M. I., and Retakh, V. S., General hypergeometric systems of equations and series of hypergeometric type. Uspekhi Mat. Nauk 47 (1992), 3–82, 235; Russian Math. Surv. 47 (1992) 1–88.
[163] Gelfand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990), 255–271.
[164] Geronimus, L. Y., Orthogonal Polynomials, American Mathematical Society, Providence, RI 1977.
[165] Glaisher, J.W. L., On elliptic functions, Messenger Math. 15 (1881–2), 81–138.
[166] Godefroy, M., La Fonction Gamma, Gauthier-Villars, Paris 1901.
[167] Gordon, W., Über den Stoss zweier Punktladungen nach der Wellenmechanik, Z. für Phys. 48 (1928), 180–191.
[168] Goursat, E., Sur l’équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. 10 (1881), 3–142.
[169] Goursat, E., Mémoire sur les fonctions hypergéométriques dordre supérieur, Ann. Sci. École Norm. Sup. 12 (1883), 261–286, 395–430.
[170] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, A., Jeffrey and D., Zwillinger, eds., Academic Press, San Diego, CA 2007.
[171] Graf, J. H., Ueber die Addition und Subtraction der Argumente bei Bessel'schen Funktionen, Math. Ann. 43 (1893), 136–144.
[172] Green, G., An Essay on the Application ofMathematical Analysis to the Theories of Electricity and Magnetism, Nottingham 1828; J. für Math. 39 (1850), 74–89.
[173] Green, G., On the motion of waves in a variable canal of small depth and width, Trans. Camb. Philos. Soc. 6 (1837), 457–462.
[174] Gromak, V. I., Laine, I., and Shimomura, S., Painlevé Differential Equations in the Complex Plane, De Gruyter, Berlin 2002.
[175] Hahn, W., Bericht über die Nullstellungen der Laguerreschen und Hermiteschen Polynome, Jahresber. Deutschen Math. Verein 44 (1934), 215–236; 45 (1935), 211.
[176] Hahn, W., Über Orthogonalpolynomen die q-Differenzgleichungen genügen, Math. Nachr. 2 (1949), 4–34.
[177] Hankel, H., Die Cylinderfunktionen erster und zweiter Art, Math. Ann. 1 (1869), 467–501.
[178] Hansen, P. A., Ermittelung der absoluten Störungen in Ellipsen von beliebiger Excentricität und Neigung, I, Schriften der Sternwarte Seeberg, Carl Gläser, Gotha 1843.
[179] Hastings, S. P. and McLeod, J. B., A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31–51.
[180] Heine, H. E., Handbuch der Kugelfunktionen, Reimer, Berlin 1878, 1881.
[181] Hermite, C., Sur une nouveau développement en série de fonctions, C. R. Acad. Sci. Paris, 58 (1864), 93–100.
[182] Higgins, J. R., Completeness and Basis Properties of Sets of Special Functions, Cambridge University Press, Cambridge 1977.
[183] Hilbert, D., Über die Diskriminanten der in endlichen abbrechenden hypergeometrischen Reihe, J. Reine Angew. Math. 103 (1885), 337–345.
[184] Hilbert, D., Über eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie. Verh. 3. Internat. Math. Kongr. Heidelberg, 1904.
[185] Hille, E., Ordinary Differential Equations in the Complex Domain, Dover, Mineola, NY 1976.
[186] Hille, E., Shohat, J., and Walsh, J. L., A Bibliography on Orthogonal Polynomials, Bulletin National Research Council, 103, National Academy of Science, Washington, DC 1940.
[187] Hobson, E. W., On a type of spherical harmonics of unrestricted degree, order, and argument, Phil. Trans. 187 (1896), 443–531.
[188] Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Chelsea, New York, NY 1965.
[189] Hochstadt, H., The Functions of Mathematical Physics, Wiley, New York, NY 1971.
[190] Hoëné-Wronski, J., Réfutation de la Théorie des Fonctions Analytiques de Lagrange, Blankenstein, Paris 1812.
[191] Horn, J., Ueber die Convergenz der hypergeometrischen Reihen zweier und dreier Vernderlichen, Math. Ann. 34 (1889), 544–600.
[192] Horn, J., Hypergeometrische Funktionen zweier Vernderlichen, Math. Ann. 105 (1931), 381–407.
[193] Ince, E. L., Ordinary Differential Equations, Dover, Mineola, NY 1956.
[194] Ismail, M. E. H., Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge 2005.
[195] Ivić, A., The Riemann Zeta-Function: Theory and Applications, Dover, Mineola, NY 2003.
[196] Iwasaki, K., Kimura, T., Shimomura, S., and Yoshida, M., From Gauss to Painlevé: A Modern Theory of Special Functions, Vieweg, Braunschweig 1991.
[197] Jacobi, C. G. J., Extraits de deux letters de M. Jacobi de l'Université de Königsberg à M. Schumacher, Schumacher Astron. Nachr. 6 (1827); Werke 1, pp. 31–36.
[198] Jacobi, C. G. J., Fundamenta Nova Theoriae Functionum Ellipticarum, Borntraeger, Königsberg 1829; Werke 1, pp. 49–239.
[199] Jacobi, C. G. J., Versuch einer Berechnung der grossen Ungleichheit des Saturns nach einer strengen Entwicklung, Schumacher Astron. Nachr. 28 (1849), 65–80, 81–94; Werke 7, pp. 145–174.
[200] Jacobi, C. G. J., Theorie der elliptischen Functionen aus den eigenschaften der Thetareihen abgeleitet, Werke 1, 1881, pp. 497–538.
[201] Jacobi, C. G. J., Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe, J. Reine Angew. Math. 56 (1859), 149–165; Werke 6, pp. 184–202.
[202] Jahnke, E. and Emde, F., Tables of Higher Functions, McGraw-Hill, New York, NY 1960.
[203] Jeffreys, H., On certain approximate solutions of linear differential equations of the second-order, Proc. London Math. Soc. 23 (1924), 428–436.
[204] Jeffreys, H., Asymptotic solutions of linear differential equations, Philos. Mag. 33 (1942), 451–456.
[205] Jimbo, M. and Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Physica D 2 (1981), 407–448.
[206] Jin, X.-S. and Wong, R., Uniform asymptotic expansions for Meixner polynomials, Constr. Approx. 14 (1998), 113–150.
[207] Jin, X.-S. and Wong, R., Asymptotic formulas for the Meixner polynomials, J. Approx. Theory 14 (1999), 281–300.
[208] Johnson, D. E. and Johnson, J. R., Mathematical Methods in Engineering and Physics: Special Functions and Boundary-Value Problems, Ronald Press, New York, NY 1965.
[209] Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen Band 1: Gewöhnliche Differentialgleichungen, Chelsea, New York, NY 1959.
[210] Kampé de Fériet, J., La Fonction Hypergéométrique, Gauthier-Villars, Paris 1937.
[211] Kampé de Fériet, J. and Appell, P. E., Fonctions Hypergéométriques et Hypersphériques, Gauthier-Villars, Paris 1926.
[212] Kanemitsu, S. and Tsukada, H., Vistas of Special Functions, World Scientific, Hackensack, NJ 2007.
[213] Kanzieper, E., Replica field theories, Painlevé transcendents, and exact correlation functions, Phys. Rev. Lett. 89 (2002), 250201.
[214] Kelvin, W. T. (Lord), On the wave produced by a single impulse in water of any depth or in a dispersive medium, Philos. Mag. 23 (1887), 252–255.
[215] Kelvin, W. T. (Lord), Presidential address to the Institute of Electrical Engineers, 1889, Math. and Phys. Papers III, pp. 484–515.
[216] Kempf, G. R., Complex Abelian Varieties and Theta Functions, Springer, Berlin 1991.
[217] Khrushchev, S., Orthogonal Polynomials and Continued Fractions from Euler's Point of View, Cambridge University Press, Cambridge 2008.
[218] Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, I, Springer, Berlin 1927.
[219] Klein, F., Vorlesungen Über die Hypergeometrische Funktion, Springer-Verlag, Berlin 1981.
[220] Knopp, K., Funktionentheorie II, 5th ed., de Gruyter, Berlin 1941.
[221] Koekoek, R.Lesky, P. A., and Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and their q-Analogues, Springer, Berlin 2010.
[222] Koelink, H. T., On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124 (1996), 887–898.
[223] Koepf, W., Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities, Vieweg, Braunschweig 1998.
[224] Kolmogorov, A. N. and Yushkevich, A. P., eds., Mathematics of the 19th Century: Function Theory According to Chebyshev, Ordinary Differential Equations, Calculus of Variations, Theory of Finite Differences, Birkhäuser, Basel 1998.
[225] Koornwinder, T., Jacobi polynomials, III. An analytical proof of the addition formula, SIAM J. Math. Anal. 6 (1975), 533–540.
[226] Koornwinder, T., Yet another proof of the addition formula for Jacobi polynomials, J. Math. Anal. Appl. 61 (1977), 136–141.
[227] Korenev, B. G., Bessel Functions and their Applications, Taylor & Francis, London 2002.
[228] Koshlyakov, N. S., On Sonine's polynomials, Messenger Math., 55 (1926), 152–160.
[229] Krall, A. M., Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhäuser, Basel 2002.
[230] Krall, H. L. and Frink, O., A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc. 65 (1948), 100–115.
[231] Kramers, H. A., Wellenmechanik und halbzahlige Quantisierung, Z. für Phys. 39 (1926), 828–840.
[232] Krawtchouk, M., Sur une généralisation des polynomes d'Hermite, C. R. Acad. Sci. Paris 189 (1929), 620–622.
[233] Kuijlaars, A., Riemann–Hilbert analysis for orthogonal polynomials, Orthogonal Polynomials and Special Functions: Leuven 2002, H. T., Koelinck and W., Van Assche, eds., Springer, Berlin 2003.
[234] Kuijlaars, A. and Van Assche, W., The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999), 167–197.
[235] Kummer, E. E., Über die hypergeometrische Reihe, J. Reine Angew. Math. 15 (1836), 127–172.
[236] Laforgia, A., Further inequalities for the gamma function, Math. Comp. 42 (1984), 597–600.
[237] Lagrange, J. L., Des oscillations d'un fil fixé par une de ses extrémités et chargés d'un nombre quelconque de poids, in the memoir Solution de différents problèmes de calcul intégral, Misc. Taur. 3 (1762–1765); Oeuvres 1, pp. 471–668.
[238] Lagrange, J. L., Sur le problème de Kepler, Hist. Acad. Sci. Berlin 25 (1769), 204–233; Oeuvres 3, pp. 113–138.
[239] Laguerre, E. N., Sur l'intégrale, Bull. Soc. Math. France 7 (1879), 72–81.
[240] Landen, J., An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom, Phil. Trans. 65 (1775), 283–289.
[241] Lang, S., Elliptic Functions, Springer, New York, NY 1987.
[242] Laplace, P. S., Théorie d'attraction des sphéroïdes et de la figure des planètes, Mém. Acad. Roy. Sci. Paris (1782, publ. 1785), 113–196.
[243] Laplace, P. S., Mémoire sur les Intégrales Définies, et leurs Applications aux Probabilités, Mém. de l'Acad. Sci., 11, Gauthier-Villars, Paris 1810–1811; Oeuvres 12, Paris 1898, pp. 357–412.
[244] Laplace, P. S., Théorie Analytique des Probabilités, 2nd ed., Oeuvres 7.
[245] Larsen, M. E., Summa Summarum, Canadian Mathematical Society, Ottawa; Peters, Wellesley, MA 2007.
[246] Lauricella, G., Sulle funzioni ipergeometriche a pi variabili, Rend. Circ. Mat. Palermo 7 (1893), 111–158.
[247] Lax, P.Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490.
[248] Lawden, D. F., Elliptic Functions and Applications, Springer, New York, NY 1989.
[249] Lebedev, N. N., Special Functions and Their Applications, Dover, Mineola, NY 1972.
[250] Legendre, A. M., Recherches sur les figures des planètes, Mém. Acad. Roy. Sci. Paris (1784, publ. 1787), 370–389.
[251] Legendre, A. M., Recherches sur l'attraction des sphéroides homogènes, Mém. Math. Phys. Acad. Sci. 10 (1785), 411–434.
[252] Legendre, A. M., Exercises de Calcul Intégrale, Courcier, Paris 1811.
[253] Legendre, A. M., Traité des Fonctions Elliptiques et des Intégrales Eulériennes, Huzard-Courcier, Paris 1825–28.
[254] Leonard, D. A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), 656–663.
[255] Levin, A. L. and Lubinsky, D. S., Orthogonal Polynomials for Exponential Weights, Springer, New York, NY 2001.
[256] Li, X. and Wong, R., On the asymptotics of the Meixner–Pollaczek polynomials and their zeros, Constr. Approx. 17 (2001), 59–90.
[257] Lin, Y. and Wong, R., Global asymptotics of the discrete Chebyshev polynomials, Asympt. Anal. 82 (2013), 39–64.
[258] Lin, Y. and Wong, R., Global asymptotics of the Hahn polynomials, Anal. Appl. (Singap.) 11 (2013), 1–47.
[259] Lin, Y. and Wong, R., Asymptotics of the Meijer G-functions, Proceedings of the Conference on Constructive Functions, 2014, D., Hardin, D., Lubinsky, and B., Simanek, eds., Contemp. Math., to appear.
[260] Liouville, J., Mémoire sur la théorie analytique de la chaleur, Math. Ann. 21 (1830–1831), 133–181.
[261] Liouville, J., Second mémoire sur de développement des fonctions ou parties de fonctions en séries dont les divers termes sont assujéties à satisfaire à une même équation de seconde ordre, contenant un paramètre variable, J. de Math. 2 (1837), 16–35.
[262] Liouville, J., Lectures published by C. W., Borchardt, J. für Math. Reine Angew. 88 (1880), 277–310.
[263] Lommel, E. C. J. von, Studien über die Bessel'schen Funktionen, Teubner, Leipzig 1868.
[264] Luke, Y. L., The Special Functions and Their Approximations, 2 vols., Academic Press, New York, NY 1969.
[265] Lützen, J., The solution of partial differential equations by separation of variables: a historical survey, Studies in the History of Mathematics, Mathematical Association of America, Washington, DC 1987.
[266] Macdonald, H. M., Note on Bessel functions, Proc. London Math. Soc. 22 (1898), 110–115.
[267] Macdonald, I. G., Symmetric Functions and Orthogonal Polynomials, American Mathematical Society, Providence, RI 1998.
[268] MacRobert, T. M., Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, Pergamon, Oxford 1967.
[269] Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, New York, NY 1949.
[270] Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, New York, NY 1966.
[271] Mainardi, F. and Pagnini, G., Salvatore Pincherle: the pioneer of theMellin–Barnes integrals, J. Comput. Appl. Math. 153 (2003), 332–342.
[272] Marcellán, F. and Álvarez-Nodarse, R., On the “Favard theorem” and its extensions, J. Comput. Appl. Math. 127 (2001), 231–254.
[273] Mathai, A. M. and Saxena, R. K., Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lect. Notes Math., 348, Springer, Berlin 1973.
[274] Mathai, A. M., Saxena, R. K., and Haubold, H. J., The H-Function, Springer, New York, NY 2010.
[275] Matsuda, M., Lectures on Algebraic Solutions of Hypergeometric Differential Equations, Kinokuniya, Tokyo 1985.
[276] McBride, E. B., Obtaining Generating Functions, Springer, New York, NY 1971.
[277] McCoy, B., Spin systems, statistical mechanics and Painlevé functions, Painlevé Transcendents, their Asymptotics and Physical Applications, NATO Adv. Sci. Inst. Ser. B: Phys., 278, Plenum, New York, NY 1992, pp. 377–391.
[278] McCoy, B., Tracy, C., and Wu, T., Painlevé functions of the third kind. Math. Phys. 18 (1977), 1058–1092.
[279] Mehler, F. G., Notiz über die Dirichlet'schen Integralausdrucke für die Kugelfuntion Pn(cos?) und über einige analoge Integralform für die CylinderfunktionJ(z), Math. Ann. 5 (1872), 141–144.
[280] Mehta, M. L., Random Matrices, Elsevier, Amsterdam 2004.
[281] Meijer, C. S., Über Whittakersche bezw. Besselsche Funktionen und deren Produkten, Nieuw Arch. Wisk. 18 (1936), 10–39.
[282] Meijer, C. S., Multiplikationstheoreme für die Funktion, Nederl. Akad. Wetensch., Proc. Ser. A 44 (1941), 1062–1070.
[283] Meijer, C. S., On the G-functions, I–VII, Nederl. Akad. Wetensch., Proc. Ser. A 49 (1946), 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072, 1165–1175.
[284] Meijer, C. S., Expansion theorems for the G-function, I–XI, Nederl. Akad. Wetensch., Proc. Ser. A 55 (1952), 369–379, 483–487; 56 (1953), 43–49, 187–193, 349–357; 57 (1954), 77–82, 83–91, 273–279; 58 (1955), 243–251, 309–314; 59 (1956), 70–82.
[285] Meixner, J., Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. 9 (1934), 6–13.
[286] Mellin, Hj., Abriss einer einheitlichen Theorie der Gamma- und der Hypergeometrischen Funktionen, Math. Ann. 68 (1910), 305–337.
[287] Miller, W., Jr., Lie Theory and Special Functions, Academic Press, New York, NY 1968.
[288] Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, Reading, MA 1977.
[289] Mittag-Leffler, G., An introduction to the theory of elliptic functions, Ann. Math. 24 (1923), 271–351.
[290] Miwa, T., Painlevé property of monodromy preserving deformation equations and the analyticity of t functions, Publ. Res. Inst. Math. Sci. 17 (1981), 703–721.
[291] Moore, G., Geometry of the string equations, Comm. Math. Phys. 133, 261–304.
[292] Motohashi, Y., Spectral Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge 1997.
[293] Mott, N. F., The solution of the wave equation for the scattering of a particle by a Coulombian centre of force, Proc. R. Soc. A 118 (1928), 542–549.
[294] Moutard, Th. F., Sur les équations différentielles linéares du second ordre, C.R. Acad. Sci. Paris 80 (1875), 729–733; J. École Polyt. 45 (1878), 1–11.
[295] Müller, C., Analysis of Spherical Symmetries in Euclidean Spaces, Springer, New York, NY 1998.
[296] Murphy, R., Treatise on Electricity, Deighton, Cambridge 1833.
[297] Murphy, R., On the inverse method of definite integrals with physical applications, Trans. Camb. Philos. Soc. 4 (1833), 353–408; 5 (1835), 315–393.
[298] Neumann, C. G., Theorie der Bessel'schen Funktionen. Ein Analogon zur Theorie der Kugelfunctionen, Teubner, Leipzig 1867.
[299] Nevai, P. G., Orthogonal Polynomials, Memoirs of the American Mathematical Society, 213, American Mathematical Society, Providence, RI 1979.
[300] Neville, E. H., Jacobian Elliptic Functions, Oxford University Press, Oxford 1951.
[301] Newman, F. W., On Г(a) especially when a is negative, Camb. Dublin Math. J. 3 (1848), 59–60.
[302] Nielsen, N., Handbuch der Theorie der Zylinderfunktionen, Teubner, Leipzig 1904.
[303] Nielsen, N., Handbuch der Theorie de Gammafunktion, Chelsea, New York, NY 1965.
[304] Nikiforov, A. F., Suslov, S. K., and Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin 1991.
[305] Nikiforov, A. F. and Uvarov, V. B., Special Functions of Mathematical Physics: A Unified Introduction with Applications, Birkhäuser, Basel 1988.
[306] Nørlund, N. E., The logarithmic solutions of the hypergeometric equation, K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 5 (1963), 1–58.
[307] Nørlund, N. E., Hypergeometric functions, Acta Math. 94 (1955), 289–349.
[308] Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer, Berlin 1990.
[309] Oberhettinger, F. and Magnus, W., Anwendung der Elliptische Funktionen in Physik und Technik, Springer, Berlin 1949.
[310] Olver, F. W. J., Second-order linear differential equations with two turning points, Philos. Trans. R. Soc. A 20 (1975), 131–174.
[311] Olver, F.W. J., Asymptotics and Special Functions, Peters, Wellesley, MA 1997.
[312] Olver, F. W. J., Lozier, D. W., Clark, C. W., and Boisvert, R. F., Digital Library of Mathematical Functions, National Institute of Standards and Technology, Gaithersburg, MD 2007.
[313] Olver, F. W. J., Lozier, D. W., Clark, C. W., and Boisvert, R. F., Handbook of Mathematical Functions, Cambridge University Press, Cambridge 2010.
[314] Painlevé, P., Gewöhnliche Differentialgleichungem: Existenz der Lösungen, Enzykl. Math. Wiss. 2 no. 2/3 (1900), 189–229.
[315] Painlevé, P., Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France 28 (1900), 201–261.
[316] Painlevé, P., Sur les équations différentielles du seconde ordre à points critiques fixes, C. R. Acad. Sci. Paris 143 (1906), 1111–1117.
[317] Painlevé, P., Oeuvres 1, 3, Centre Nationale de la Recherche Scientifique, Paris 1972, 1975.
[318] Pan, J. H. and Wong, R., Uniform asymptotic expansions for the discrete Chebyshev polynomials, Stud. Appl. Math. 128 (2011), 337–384.
[319] Patterson, S. J., An Introduction to the Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge 1988.
[320] Perron, O., Die Lehre von den Kettenbruchen, 2nd ed., Teubner, Leipzig 1929.
[321] Penson, K. A., Blasiak, P., Duchamp, G. H. E., Horzela, A., and Solomon, A. I., On certain non-unique solutions of the Stieltjes moment problem, Disc. Math. Theor. Comp. Sci. 12 (2010), 295–306.
[322] Petiau, G., La Théorie des Fonctions de Bessel Exposée en vue de ses Applications à la Physique Mathématique, Centre National de la Recherche Scientifique, Paris 1955.
[323] Petkovšek, M., Wilf, H. S., and Zeilberger, D., A = B, Peters, Wellesley, MA 1996.
[324] Pfaff, J. F., Nova disquisitio de integratione aequationes differentio-differentialis, Disquisitiones Analyticae, Helmstadt 1797.
[325] Picard, É., Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles, Bull. Sci. Math. 2 12 (1888), 148–156.
[326] Pincherle, S., Sopra una trasformazione delle equazioni differenziali lineari in equazioni lineari alle differenze, e viceversa, R. Ist. Lomb. Sci. Lett. Rend. 19 (1886), 559–562.
[327] Pincherle, S.Sulle funzioni ipergeometriche generalizzate, Atti R. Accad. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 4 (1888) 694–700, 792–799.
[328] Plancherel, M. and Rotach, W., Sur les valeurs asymptotiques des polynomes d'Hermite Hn(x) = (1)nex2/2dn(ex2/2)/dxn, Comm. Math. Helvetici 1 (1929), 227–254.
[329] Plemelj, J., Riemannsche Formenscharen mit gegebener Monodromiegruppe. Monatsh. Math. Physik 19 (1908), 211–246.
[330] Pochhammer, L., Über hypergeometrische Functionen n-ter Ordnung, J. Reine Angew. Math. 71 (1870), 316–352.
[331] Poisson, S. D., Mémoire sur la distribution de la chaleur dans les corps solides, J. École Roy. Polyt. 19 (1823), 1–162, 249–403.
[332] Polishchuk, A., Abelian Varieties, Theta Functions and the Fourier Transform, Cambridge University Press, Cambridge 2003.
[333] Pollaczek, F., Sur une généralisation des polynômes de Legendre, C. R. Acad. Sci. Paris 228 (1949), 1363–1365.
[334] Prasolov, V. and Solovyev, Yu., Elliptic Functions and Elliptic Integrals, American Mathematical Society, Providence, RI 1997.
[335] Prosser, R. T., On the Kummer solutions of the hypergeometric equation, Amer. Math. Monthly 101 (1994), 535–543.
[336] Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, vols. 1–5. Gordon and Breach, New York, NY1986–2004.
[337] Qiu, W.-Y. and Wong, R., Asymptotic expansion of the Krawtchouk polynomials and their zeros, Comput. Methods Funct. Theory 4 (2004), 189–226.
[338] Qiu, W.-Y. and Wong, R., Global asymptotic expansions of the Laguerre polynomials – a Riemann–Hilbert approach, Numer. Algorithms 49 (2008), 331–372.
[339] Rainville, E. D., Special Functions, Chelsea, New York, NY 1971.
[340] Remmert, R., Wielandt's theorem about the Г-function, Amer. Math. Monthly 103 (1996), 214–220.
[341] Riccati, J. F. (Count), Animadversiones in aequationes differentiales secundi gradus, Acta Erud. Publ. Lipsiae Suppl. 8 (1724), 66–73.
[342] Riemann, B., Zwei allgemeine Lehrsätze für lineäre Differentialgleichungen mit algebraischen Koefficienten, Ges. Math. Werke, pp. 357–369.
[343] Riemann, B., Beiträge zur Theorie der durch Gauss'she Reihe F(a, ß, y, x) darstellbaren Funktionen, Kön. Ges. Wiss. Gött. 7 (1857), 1–24; Werke, pp. 67–90.
[344] Robin, L., Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vols. I–III, Gauthier-Villars, Paris1957–1959.
[345] Rodrigues, O., MÉmoire sur l'attraction des sphÉroides, Corr. École Roy. Polyt. 3 (1816), 361–385.
[346] Routh., E., On some properties of certain solutions of a differential equation of the second-order, Proc. London Math. Soc. 16 (1885), 245–261.
[347] Roy, R., Sources in the Development of Mathematics. Series and Products from the Fifteenth to the Twenty-first Century, Cambridge University Press, Cambridge 2011.
[348] Russell, A., The effective resistance and inductance of a concentric main, and methods of computing the Ber and Bei and allied functions, Philos. Mag. 17 (1909), 524–552.
[349] Sachdev, P. L., A Compendium on Nonlinear Ordinary Differential Equations, Wiley, New York, NY 1997.
[350] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields, Springer, New York, NY 1997.
[351] Sansone, G., Orthogonal Functions, Dover, Mineola, NY 1991.
[352] Šapiro, R. L., Special functions: related to representations of the group SU(n) of class I with respect to SU(n-1) n = 3 (in Russian), Izv. Vysš. Učebn. Zaved. Mat. 4 (1968), 97–107.
[353] Sasvari, Z., An elementary proof of Binet's formula for the Gamma function, Amer. Math. Monthly 106 (1999), 156–158.
[354] Schläfli, L., Sull'uso delle linee lungo le quali il valore assoluto di una funzione è constante, Annali Mat. 2 (1875), 1–20.
[355] Schläfli, L., Ueber die zwei Heine'schen Kugelfunktionen, Bern 1881.
[356] Schlömilch, O., Analytische Studien, Engelmann, Leipzig 1848.
[357] Schlömilch, O., Ueber die Bessel'schen Funktion, Z. Math. Phys. 2 (1857), 137–165.
[358] Schmidt, H., Über Existenz und Darstellung impliziter Funktionen bei singul ären Anfangswerten, Math. Z. 43 (1937/38), 533–556.
[359] Schwarz, H. A., Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292–335.
[360] Seaborn, J. B., Hypergeometric Functions and Their Applications, Springer, New York, NY 1991.
[361] Seaton, M. J., Coulomb functions for attractive and repulsive potentials and for positive and negative energies, Comput. Phys. Comm. 146 (2002), 225–249.
[362] Selberg, A., Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 24 (1944), 159–171.
[363] Sharapudinov, I. I., Asymptotic properties and weighted estimates for orthogonal Chebyshev–Hahn polynomials, Mat. Sb. 182 (1991), 408–420; Math. USSR Sb. 72 (1992), 387–401.
[364] Sherman, J., On the numerators of the Stieltjes continued fractions, Trans. Amer. Math. Soc. 35 (1933) 64–87.
[365] Shohat, J., The relation of classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math. 58 (1936), 453–464.
[366] Simon, B., The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), 82–203.
[367] Simon, B., Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, American Mathematical Society, Providence, RI 2005.
[368] Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, New York, NY 1960.
[369] Slater, L. J., Generalized Hypergeometric Functions, Cambridge University Press, Cambridge 1966.
[370] Slavyanov, S. Yu. and Lay, W., Special Functions: A Unified Theory Based on Singularities, Oxford University Press, Oxford 2000.
[371] Sneddon, I. N., Special Functions of Mathematical Physics and Chemistry, 3rd ed., Longman, London 1980.
[372] Sokhotskii, Y. W., On definite integrals and functions used in series expansions, Thesis, St. Petersburg 1873.
[373] Sommerfeld, A. J. W., Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), 317–374.
[374] Sonine, N. J., Recherches sur les fonctions cylindriques et le développment des fonctions continues en séries, Math. Ann. 16 (1880), 1–80.
[375] Stahl, H. and Totik, V., General Orthogonal Polynomials, Cambridge University Press, Cambridge 1992.
[376] Stanton, D., Orthogonal polynomials and Chevalley groups, Special Functions: Group Theoretic Aspects and Applications, R., Askey, T., Koornwinder, and W., Schempp, eds., Reidel, New York, NY 1984, pp. 87–128.
[377] Steeb, W.-H. and Euler, N., Nonlinear Evolution Equations and Painlevé Test, World Scientific Publishing, Singapore 1988.
[378] Steklov, V. A., Sur les expressions asymptotiques de certaines fonctions, définies par les équations différentielles linéaires de seconde ordre, et leurs applications au problème du développement d'une fonction arbitraire en séries procédant suivant les-dites fonctions, Comm. Soc. Math. Kharkhow 10 (1907), 197–200.
[379] Sternberg, W. J. and Smith, T. L., The Theory of Potential and Spherical Harmonics, University of Toronto Press, Toronto 1944.
[380] Stieltjes, T. J., Quelques recherches sur la théorie des quadratures dites mÉcaniques, Ann. Sci. École Norm. Sup. 1 (1884), 409–426.
[381] Stieltjes, T. J., Sur quelques théorèmes d'algèbre, C. R. Acad. Sci. Paris 100 (1885), 439–440; Oeuvres 1, pp. 440–441.
[382] Stieltjes, T. J., Sur les polynômes de Jacobi, C. R. Acad. Sci. Paris 100 (1885), 620–622; Oeuvres 1, pp. 442–444.
[383] Stieltjes, T. J., Sur les polynômes de Legendre, Ann. Fac. Sci. Toulouse 4 (1890), G1–G17; Oeuvres 2, pp. 236–252.
[384] Stieltjes, T. J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 9 (1894), J1–122; 9 (1895), A1–10.
[385] Stillwell, J., Mathematics and its History, Springer, New York, NY 2002.
[386] Stirling, J., Methodus Differentialis, London 1730.
[387] Stokes, G. G., On the numerical calculation of a class of definite integrals and infinite series, Trans. Camb. Philos. Soc. 9 (1850), 166–187.
[388] Stokes, G. G., On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Camb. Philos. Soc. 10 (1857), 105–128.
[389] Stone, M. H., Linear Transformations in Hilbert Space and their Applications to Analysis, American Mathematical Society, Providence, RI 1932.
[390] Sturm, J. C. F., Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186.
[391] Szegő, G., On an inequality of P. Turán regarding Legendre polynomials, Bull. Amer. Math. Soc. 54 (1949), 401–405.
[392] Szegő, G., Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI 1975.
[393] Szegő, G., An outline of the history of orthogonal polynomials, Orthogonal Expansions and their Continuous Analogues, Southern Illinois University Press, Carbondale, IL 1968, pp. 3–11.
[394] Talman, J. D., Special Functions: A Group Theoretic Approach, Benjamin, New York, NY 1968.
[395] Tannery, J. and Molk, J., ÉlÉments de la ThÉorie des Fonctions Elliptiques, vols. 1–4, Gauthier-Villars, Paris1893–1902; Chelsea, New York, NY 1972.
[396] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572.
[397] Temme, N. M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, NY 1996.
[398] Thomae, J., Ueber die höheren hypergeometrische Reihen, insbesondere über die ReiheMath. Ann. 2 (1870), 427–441.
[399] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd. ed., Oxford University Press, New York, NY 1986.
[400] Tracy, C. A. and Widom, H., Painlevé functions in statistical physics, Publ. RIMS Kyoto Univ. 47 (2011), 361–374.
[401] Tricomi, F., Serie Ortogonale di Funzioni, S. I. E. Istituto Editoriale Gheroni, Torino 1948.
[402] Tricomi, F., Funzioni Ipergeometrichi Confluenti, Ed. Cremonese, Rome 1954.
[403] Tricomi, F., Funzioni Ellittiche, Zanichelli, Bologna 1951.
[404] Truesdell, C., An Essay Toward a Unified Theory of Special Functions Based upon the Functional Equation (∂/∂z)F(z,a) = F(z,a + 1), Annals of Mathematics Studies, 18, Princeton University Press, Princeton, NJ 1948.
[405] Turán, P., On the zeros of the polynomials of Legendre, Čas. Pešt. Mat. Fys. 75 (1950), 113–122.
[406] Uspensky, J. V., On the development of arbitrary functions in series of orthogonal polynomials, Ann. Math. 28 (1927), 563–619.
[407] Van Assche, W., Asymptotics for Orthogonal Polynomials, Lect. Notes Math., 1265, Springer, Berlin 1987.
[408] van der Corput, J. G., Asymptotic Expansions. I–III, Department of Mathematics, University of California, Berkeley, CA1954–1955.
[409] van der Laan, C. G. and Temme, N. M., Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-like Functions, Centrum voor Wiskunde en Informatica, Amsterdam 1984.
[410] Vandermonde, A., Mémoire sur des irrationelles de différens ordres avec une application au cercle, Mém. Acad. Roy. Sci. Paris (1772), 489–498.
[411] Varadarajan, V. S., Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. 33 (1996), 1–42.
[412] Varchenko, A., Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, World Scientific Publishing, River Edge, NJ 1995.
[413] Varchenko, A., Special Functions, KZ Type Equations, and Representation Theory, American Mathematical Society, Providence, RI 2003.
[414] Vilenkin, N. Ja., Special Functions and the Theory of Group Representations, American Mathematical Society, Providence, RI 1968.
[415] Vilenkin, N. Ja. and Klimyk, A. U., Representation of Lie Groups and Special Functions, 3 vols., Kluwer, Dordrecht1991–1993.
[416] Walker, P. L., Elliptic Functions, Wiley, Chichester 1996.
[417] Wallis, J., Arithmetica Infinitorum, Oxford 1656.
[418] Wang, X.-S. and Wong, R., Asymptotics of orthogonal polynomials via recurrence relations, Anal. Appl. 10 (2012), 215–235.
[419] Wang, X.-S. and Wong, R., Asymptotics of the Racah polynomials with varying parameters, J. Math. Anal. Appl., to appear.
[420] Wang, Z. and Wong, R., Asymptotic expansions for second-order linear difference equations with a turning point, Numer. Math. 94 (2003), 147–194.
[421] Wang, Z. and Wong, R., Linear difference equations with transition points, Math. Comp. 74 (2005), 629–653.
[422] Wang, Z.-X. and Guo, D. R., Special Functions, World Scientific, Teaneck, NJ 1989.
[423] Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, Dover, Mineola, NY 1987.
[424] Watson, G. N., Bessel functions and Kapteyn series, Proc. London Math. Soc. 16 (1917), 150–174.
[425] Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge 1995.
[426] Wawrzyńczyk, A., Group Representations and Special Functions, Reidel, Dordrecht 1984.
[427] Weber, H., Über die Integration der partiellen Differentialgleichung: Math. Ann. 1 (1869), 1–36.
[428] Weber, H., Über eine Darstellung willkürlicher Funktionen durch Bessel'sche Funktionen, Math. Ann. 6 (1873), 146–161.
[429] Weber, M. and Erdélyi, A., On the finite difference analogue of Rodrigues' formula, Amer. Math. Monthly 59 (1952), 163–168.
[430] Weierstrass, K. L., Formeln und Lehrsätze zum Gebrauch der Elliptische Funktionen, Kaestner, Göttingen1883–1885.
[431] Weierstrass, K. L., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsber. Königl. Preuss. Akad. Wissensch. II (1885), 633–639, 789–805.
[432] Wentzel, G., Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Z. für Phys. 38 (1926), 518–529.
[433] Whitehead, C. S., On the functions berx, beix, kerx, and keix, Quarterly J. 42 (1909), 316–342.
[434] Whittaker, E. T., An expression of certain known functions as generalized hypergeometric functions, Bull. Amer. Math. Soc. 10 (1903), 125–134.
[435] Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, Cambridge University Press, Cambridge 1969.
[436] Wiles, A., Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443–551.
[437] Wilf, H. S. and Zeilberger, D., An algebraic proof theory for geometric (ordinary and “q”) multisum/integral identities, Invent. Math. 108 (1992), 575–633.
[438] Wilson, J. A., Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), 690–701.
[439] Wilson, J. A., Asymptotics for the 4F3 polynomials, J. Approx. Theory 66 (1991), 58–71.
[440] Wimp, J., Recursion formulae for hypergeometric functions, Math. Comp. 22 (1968), 363–373.
[441] Wintner, A., Spektraltheorie der Unendlichen Matrizen, Einführung in den Analytischen Apparat der Quantenmechanik, Hitzel, Leipzig 1929.
[442] Wolfram, S., Festschrift for Oleg Marichev, http://www.stephenwolfram.com/publications/history-future-special-functions/intro.html
[443] Wong, R., Asymptotic Approximations of Integrals, Academic Press, Boston, MA 1989; SIAM, Philadelphia, PA 2001.
[444] Wong, R. and Li, H., Asymptotic expansions for second-order linear difference equations, J. Comp. Appl. Math. 41 (1992), 65–94.
[445] Wong, R. and Zhang, L., Global asymptotics of Hermite polynomials via Riemann–Hilbert approach, Discrete Cont. Dyn. Syst. B 7 (2007), 661–682.
[446] Yost, F. L., Wheeler, J. A., and Breit, G., Coulomb wave functions in repulsive fields, Phys. Rev. 49 (1936), 174–189.
[447] Zhang, S. and Jin, J., Computation of Special Functions, Wiley, New York, NY 1996.
[448] Zhou, X., The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20 (1989), 966–986.
[449] Zwillinger, D., Handbook of Differential Equations, Wiley, New York, NY 1998.
