publication . Preprint . 2012

An asymptotic formula of the divergent bilateral basic hypergeometric series

Morita, Takeshi;
Open Access English
  • Published: 07 May 2012
Abstract
We show an asymptotic formula of the divergent bilateral basic hypergeometric series ${}_1\psi_0 (a;-;q,\cdot)$ with using the $q$-Borel-Laplace method. We also give the limit $q\to 1-0$ of our asymptotic formula.
Subjects
free text keywords: Mathematics - Classical Analysis and ODEs, 33D15, 34M40, 39A13
Related Organizations
Download from
16 references, page 1 of 2

[1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999.

[2] G. D. Birkhoff, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Am. Acad. Arts and Sciences, 49 (1914), 521 − 568.

[3] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed, Cambridge, 2004.

[4] G. H. Hardy, Ramanujan, Cambridge University Press, Cambridge; reprinted by Chelsea, New York, 1978.

[5] Heine. E. Untersuchungen u¨ber die Reihe..., J. reine angew. Math. 34, 285-328.

[6] M. E. Horn, Bilateral binomial theorem, SIAM Problem 03-001 (2003). [OpenAIRE]

[7] T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), 445-461.

[8] T. Morita, A connection formula of the Hahn-Exton q-Bessel Function, SIGMA,7 (2011), 115, 11pp.

[9] T. Morita, A connection formula of the q-confluent hypergeometric function, arXiv:1105.5770.

[10] T. Morita, On local solutions of the Ramanujan equation and their connection formulae, arXiv:1203.3404.

[11] J. Sauloy, Algebraic construction of the Stokes sheaf for irregular linear q-difference equations, arXiv:math/0409393

[12] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. MR0201688 (34:1570)

[13] G. N. Watson, The continuation of functions defined by generalized hypergeometric series, Trans. Camb. Phil. Soc. 21 (1910), 281-299.

[14] C. Zhang, Remarks on some basic hypergeometric series, in “Theory and Applications of Special Functions”, Springer (2005), 479-491.

[15] C. Zhang, Sur les fonctions q-Bessel de Jackson, J. Approx. Theory, 122 (2003), 208-223.

16 references, page 1 of 2
Any information missing or wrong?Report an Issue