Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-29T13:00:05.381Z Has data issue: false hasContentIssue false
  • English
  • Français

Bilinear and bilateral generating functions of generalized polynomials

Published online by Cambridge University Press:  17 February 2009

Ch. Wali Mohammad
Ch. Wali Mohammad
Affiliation:
Faculty of Engineering and Technology J. M. I. University, New Delhi, INDIA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper contains mainly three theorems involving generating functions expressed in terms of single and double Laplace and beta integrals. The theorems, in turn, yield, as special cases, a number of bilinear and bilateral generating functions of generalized functions particularly general double and triple hypergeometric series. One variable special cases of the generalized functions are important in several applied problems.

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 2 , October 1997 , pp. 257 - 270
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Appell, P. and Fériet's, J. Kampé dé, Fonctions hypergèometriques et hypersphériques: Polynomes d'Hermite (Gauthier-Villars et cie, Paris, 1926).Google Scholar
[2]Bateman, H., “A generalization of Legendre polynomial”, Proc. Lond. Math. Soc. Series (2) 3 (1905) 111–123.CrossRefGoogle Scholar
[3]Bedient, P. E., “Polynomials related to Appell's functions of two variables”, Ph. D. Thesis, Univirsity of Michigan, 1959.Google Scholar
[4]Brafman, F., “Generating function of Jacobi and related polynomials”, Proc. Amer. Math. Soc. 2 (1951) 942–949.CrossRefGoogle Scholar
[5]Burchall, J. L. and Chaundy, T. W., “Expansions of Appell's double hypergeometric functions (II)”, Quart. J. Math. Oxford Ser. 12 (1941) 112–128.CrossRefGoogle Scholar
[6]Carlitz, L., “A bilinear generating function for the Jacobi polynomials”, Bull. Un. Math. Ital. (3) 18 (1963) 87–89.Google Scholar
[7]Mohd, Chaudhary Wali. and Qureshi, M. I., “Expansion formulae for general triple hypergeometric series”, J. Austral. Math. Soc. Ser. B 27 (1986) 376–385.CrossRefGoogle Scholar
[8]Chhabra, S. P. and Rusia, K. C., “A transformation formula for a general hypergeometric function of three variables”, Jnänäbha 9/10 (1980) 155–159.Google Scholar
[9]Deshpande, V. L., “Certain formulas associated with hypergeometric functions of three variables”, Pure and Applied Mathematics Science 14 (1981) 39–45.Google Scholar
[10]Erdélyi, A., Higher Transcendental Functions Vol. I (Bateman manuscript Project McGraw-Hill Book Co. Inc., New York, Toronto and London, 1953).Google Scholar
[11]Erdélyi, A., Higher Transcendental Functions Vol. III (Bateman manuscript Project McGraw-Hill Book Co. Inc., New York, Toronto and London, 1955).Google Scholar
[12]Exton, H., Multiple hypergeometric functions and applications (John Wiley and Sons (Halsted Press), New York, Ellis Horwood, Chichester, 1976).Google Scholar
[13]Halim, N. A. and Al-Salam, W. A., “Double Euler transformations of certain hypergeometric functions”, Duke Math. J. 30 (1963) 51–62.Google Scholar
[14]Jain, R. N., “The confluent hygergeometric functions of three variables”, Proc. Nat. Acad. Sci. India Sect. A 36 (2) (1966) 395–408.Google Scholar
[15]Khandekar, P. R., “On a generalization of Rice polynomial”, Proc. Nat. Acad. Sci. India Sect. A 34 (2) (1964) 157–162.Google Scholar
[16]Lauricella, G., “Sulle funzioni ipergeometriche a piú variabili”, Rend. Circ. Mat. Palermo 7 (1893) 111–158.CrossRefGoogle Scholar
[17]Manocha, H. L., “Some bilinear generating functions for Jacobi polynomials”, Proc. Camb. Philos. Soc. 63 (1967) 457–459.CrossRefGoogle Scholar
[18]Manocha, H. L., “Some formulae for generalized Rice polynomials”, Proc. Camb. Philos. Soc. 64 (1968) 431–434.CrossRefGoogle Scholar
[19]Manocha, H. L. and Sharma, B. L., “Infinite series of hypergeometric functions”, Ann. Soc. Sci. Bruxelles Ser. l 80 (1966) 73–86.Google Scholar
[20]Mathur, B. L., “On some results involving Lauricella functions”, Bull. Cal. Math. Soc. 70 (1978) 221–227.Google Scholar
[21]Munot, P. C., Mathur, B. L. and Kushwaha, R. S., “On generating functions for classical polynomials”, Proc. Nat. Acad. Sci. India Sect. A 45 (1975) 187–192.Google Scholar
[22]Rainville, E. D., Special functions, (MacMillan Co., New York, 1960), reprinted by Chelsea Publ. Co., Bronx, New York, 1971.Google Scholar
[23]Saran, S., “Hypergeometric functions of three variables”, Ganita 5 (1954) 71–91, Corrigendum. Ibid 7 (1956), 65.Google Scholar
[24]Saran, S., “A general theorem for bilinear generating functions”, Pacific J. Math. 35 (1970) 783–786.CrossRefGoogle Scholar
[25]Saran, S., “Theorems on bilinear generating functions”, Indian J. Pure Appl. Math. 3 (1) (1972) 12–20.Google Scholar
[26]Sharma, B. L., “Some theorems for Appell's functions”, Proc. Camb. Philos. Soc. 67 (1970) 613–618.CrossRefGoogle Scholar
[27]Sharma, B. L. and Mittal, K. C., “Some new generating functions for Jacobi polynomials”, Proc. Camb. Philos. Soc. 54 (1968) 691–694.CrossRefGoogle Scholar
[28]Srivastava, H. M., “Hypergeometric functions of three variables”, Ganita 15 (2) (1964) 97–108.Google Scholar
[29]Srivastava, H. M., “Generalized Neumann expansions involving hypergeometric functions”, Proc. Camb. Philos. Soc. 63 (1967) 428–429.CrossRefGoogle Scholar
[30]Srivastava, H. M., “Some integrals representing triple hypergeometric functions”, Rend. Circ. Mat. Palermo (2) 16 (1–3) (1967) 99–115.CrossRefGoogle Scholar
[31]Srivastava, H. M. and Manocha, H. L., A treatise on generating functions (Hasted Press (Ellis Horwood, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984).Google Scholar