𝑞-extension of the 𝑝-adic gamma function. II

Koblitz, Neal

doi:10.1090/S0002-9947-1982-0664032-4

$q$-extension of the $p$-adic gamma function. II
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Trans. Amer. Math. Soc. 273 (1982), 111-129 Request permission

Abstract:

Taylor series and asymptotic expansions are developed for $q$-extensions of the $p$-adic psi (derivative of log-gamma) function "twisted" by roots of unity. Connections with $p$-adic $L$-functions and $q$-expansions of Eisenstein series are discussed. The $p$-adic series are compared with the analogous classical expansions.

References

George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
Emil Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964. Translated by Michael Butler. MR 0165148
Richard Askey, The $q$-gamma and $q$-beta functions, Applicable Anal. 8 (1978/79), no. 2, 125–141. MR 523950, DOI 10.1080/00036817808839221
Richard Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), no. 5, 346–359. MR 567718, DOI 10.2307/2321202
Jack Diamond, The $p$-adic log gamma function and $p$-adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321–337. MR 498503, DOI 10.1090/S0002-9947-1977-0498503-9
Jack Diamond, On the values of $p$-adic $L$-functions at positive integers, Acta Arith. 35 (1979), no. 3, 223–237. MR 550294, DOI 10.4064/aa-35-3-223-237
Kenkichi Iwasawa, Lectures on $p$-adic $L$-functions, Annals of Mathematics Studies, No. 74, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0360526, DOI 10.1515/9781400881703
Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 69–190. MR 0447119
Nicholas M. Katz, $p$-adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104 (1976), no. 3, 459–571. MR 506271, DOI 10.2307/1970966
Nicholas M. Katz, The Eisenstein measure and $p$-adic interpolation, Amer. J. Math. 99 (1977), no. 2, 238–311. MR 485797, DOI 10.2307/2373821

-adic

-functions for CM fields

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Nicholas M. Katz, $p$-adic $L$-functions, Serre-Tate local moduli, and ratios of solutions of differential equations, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 365–371. MR 562628
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. MR 0466081, DOI 10.1007/978-1-4684-0047-2
Neal Koblitz, A new proof of certain formulas for $p$-adic $L$-functions, Duke Math. J. 46 (1979), no. 2, 455–468. MR 534062
Neal Koblitz, $p$-adic analysis: a short course on recent work, London Mathematical Society Lecture Note Series, vol. 46, Cambridge University Press, Cambridge-New York, 1980. MR 591682, DOI 10.1017/CBO9780511526107
Neal Koblitz, $q$-extension of the $p$-adic gamma function, Trans. Amer. Math. Soc. 260 (1980), no. 2, 449–457. MR 574791, DOI 10.1090/S0002-9947-1980-0574791-5
M. Krasner, Rapport sur le prolongement analytique dans les corps valués complets par la méthode des éléments analytiques quasi-connexes, Table Ronde d’Analyse non archimédienne (Paris, 1972) Bull. Soc. Math. France, Mém. No. 39–40, Soc. Math. France, Paris, 1974, pp. 131–254 (French). MR 0385168, DOI 10.24033/msmf.166
Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
Serge Lang, Introduction to modular forms, Grundlehren der Mathematischen Wissenschaften, No. 222, Springer-Verlag, Berlin-New York, 1976. MR 0429740
Serge Lang, Cyclotomic fields, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. MR 0485768, DOI 10.1007/978-1-4612-9945-5
Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759