A new elliptic interpolation formula via the (𝑓,𝑔)-inversion

Wang, Jin

doi:10.1090/proc/15043

A new elliptic interpolation formula via the $(f,g)$-inversion
HTML articles powered by AMS MathViewer

by Jin Wang PDF

Proc. Amer. Math. Soc. 148 (2020), 3457-3471 Request permission

Abstract:

In the present paper, we establish an elliptic interpolation formula with the help of the $(f,g)$-inversion formula. As applications, some basic theta function identities are presented, including an equivalent algebraic form of Weierstrass’ theta identity.

References

Hina Manoj Arora and Swadesh Kumar Sahoo, Interpolation on Gauss hypergeometric functions with an application, Involve 11 (2018), no. 4, 625–641. MR 3778916, DOI 10.2140/involve.2018.11.625
Sandy H. L. Chen and Amy M. Fu, A $2n$-point interpolation formula with its applications to $q$-identities, Discrete Math. 311 (2011), no. 16, 1793–1802. MR 2806043, DOI 10.1016/j.disc.2011.04.019
Sandy H. L. Chen and Amy M. Fu, A $4n$-point elliptic interpolation formula and its applications, SIAM J. Discrete Math. 31 (2017), no. 2, 758–765. MR 3634985, DOI 10.1137/120875521
Chu Wenchang, Shalosh B. Ekhad, and Robin J. Chapman, Problems and Solutions: Solutions: 10226, Amer. Math. Monthly 103 (1996), no. 2, 175–177. MR 1542800, DOI 10.2307/2975119
Tianzhu Dai and Xinrong Ma, A bivariate circular summation formula for cubic theta functions and its implications, J. Ramanujan Math. Soc. 26 (2011), no. 3, 237–259. MR 2865818
G. P. Egorychev, Integral representation and the computation of combinatorial sums, Translations of Mathematical Monographs, vol. 59, American Mathematical Society, Providence, RI, 1984. Translated from the Russian by H. H. McFadden; Translation edited by Lev J. Leifman. MR 736151, DOI 10.1090/mmono/059
Amy M. Fu and Alain Lascoux, A Newton type rational interpolation formula, Adv. in Appl. Math. 41 (2008), no. 3, 452–458. MR 2449602, DOI 10.1016/j.aam.2008.01.003
George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
Mourad E. H. Ismail, The Askey-Wilson operator and summation theorems, Mathematical analysis, wavelets, and signal processing (Cairo, 1994) Contemp. Math., vol. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 171–178. MR 1354852, DOI 10.1090/conm/190/02300
Mourad E. H. Ismail and Dennis Stanton, Applications of $q$-Taylor theorems, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 259–272. MR 1985698, DOI 10.1016/S0377-0427(02)00644-1
Mourad E. H. Ismail and Dennis Stanton, Some combinatorial and analytical identities, Ann. Comb. 16 (2012), no. 4, 755–771. MR 3000443, DOI 10.1007/s00026-012-0158-1
Mourad E. H. Ismail and Mizan Rahman, Inverse operators, $q$-fractional integrals, and $q$-Bernoulli polynomials, J. Approx. Theory 114 (2002), no. 2, 269–307. MR 1883409, DOI 10.1006/jath.2001.3644
Tom H. Koornwinder, On the equivalence of two fundamental theta identities, Anal. Appl. (Singap.) 12 (2014), no. 6, 711–725. MR 3271026, DOI 10.1142/S0219530514500559
Alain Lascoux, Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, vol. 99, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. MR 2017492, DOI 10.1090/cbms/099
Zhi-Guo Liu, An addition formula for the Jacobian theta function and its applications, Adv. Math. 212 (2007), no. 1, 389–406. MR 2319774, DOI 10.1016/j.aim.2006.10.005
X. R. Ma, An extension of Warnaar’s matrix inversion, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3179–3189. MR 2160179, DOI 10.1090/S0002-9939-05-07912-8
Xinrong Ma, The $(f,g)$-inversion formula and its applications: the $(f,g)$-summation formula, Adv. in Appl. Math. 38 (2007), no. 2, 227–257. MR 2290811, DOI 10.1016/j.aam.2005.06.006
Michael J. Schlosser, A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator, Special functions and orthogonal polynomials, Contemp. Math., vol. 471, Amer. Math. Soc., Providence, RI, 2008, pp. 175–186. MR 2367418, DOI 10.1090/conm/471/09213
Michael J. Schlosser and Meesue Yoo, Elliptic hypergeometric summations by Taylor series expansion and interpolation, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 039, 21. MR 3488171, DOI 10.3842/SIGMA.2016.039
J. Wang, Studies on some problems of combinatorial inversions and $q$-series transformations, Ph.D. Thesis, Soochow University, P. R. China, 2019.