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LINEAR INDEPENDENCE OF VALUES OF THE q-EXPONENTIAL AND RELATED FUNCTIONS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  23 October 2023

ANUP B. DIXIT Open the ORCID record for ANUP B. DIXIT [Opens in a new window] ,
VEEKESH KUMAR Open the ORCID record for VEEKESH KUMAR [Opens in a new window]  and
SIDDHI S. PATHAK Open the ORCID record for SIDDHI S. PATHAK [Opens in a new window]
ANUP B. DIXIT
Affiliation:
Institute of Mathematical Sciences (HBNI), CIT Campus Taramani, Chennai, Tamil Nadu 600113, India e-mail: anupdixit@imsc.res.in
VEEKESH KUMAR*
Affiliation:
Department of Mathematics, Indian Institute of Technology, Dharwad, Karnataka 580011, India
SIDDHI S. PATHAK
Affiliation:
Chennai Mathematical Institute, H-1 SIPCOT IT Park, Siruseri, Kelambakkam, Tamil Nadu 603103, India e-mail: siddhi@cmi.ac.in
*
e-mail: veekeshk@iitdh.ac.in
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Abstract

We establish the linear independence of values of the q-analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q-series.

Keywords

q-exponential functionq-logarithm functionPV number

MSC classification

Primary: 11J72: Irrationality; linear independence over a field
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Research of the first and the third authors was partially supported by the INSPIRE Faculty Fellowship.

References

Amou, M., Matala-Aho, T. and Väänänen, K., ‘On Siegel–Shidlovskii’s theory for $q$ -difference equations’, Acta Arith. 127 (2007), 309–335.10.4064/aa127-4-2CrossRefGoogle Scholar
Corvaja, P. and Zannier, U., Applications of Diophantine Approximation to Integral Points and Transcendence, Cambridge Tracts in Mathematics, 212 (Cambridge University Press, Cambridge, 2018).10.1017/9781108348096CrossRefGoogle Scholar
Duverney, D., ‘Irrationalité de la somme des inverses de la suite de Fibonacci’, Elem. Math. 52 (1997), 31–36.10.1007/s000170050008CrossRefGoogle Scholar
Erdős, P., ‘On arithmetical properties of Lambert series’, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66.Google Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edn, Encyclopedia of Mathematics and Its Applications, 96 (Cambridge University Press, Cambridge, 2004).10.1017/CBO9780511526251CrossRefGoogle Scholar
Krattenthaler, C., Rivoal, T. and Zudilin, W., ‘Series hypergeometriques basiques, $q$ -analogues de valeurs de la fonction zeta et series d’Eisenstein’, J. Inst. Math. Jussieu 5(1) (2006), 53–79.10.1017/S1474748005000149CrossRefGoogle Scholar
Murty, R., ‘The Fibonacci zeta-function’, in: Automorphic Representations and $L$ -functions, Tata Institute of Fundamental Research Conference Proceedings, 18 (eds. Prasad, D., Rajan, C. S., Sankaranarayanan, A. and Sengupta, J.) (Hindustan Book Agency, New Delhi, India, 2013).Google Scholar
Pisot, C., ‘Kriterium für die algebraischen Zahlen’, Math. Z. 48 (1942), 293–323.10.1007/BF01180020CrossRefGoogle Scholar
Tachiya, Y., ‘Irrationality of certain Lambert series’, Tokyo J. Math. 27(1) (2004), 75–85.10.3836/tjm/1244208475CrossRefGoogle Scholar
Zudilin, W., ‘On the irrationality of generalized $q$ -logarithm’, Res. Number Theory 2 (2016), Article no. 15.10.1007/s40993-016-0042-xCrossRefGoogle Scholar