An inverse factorial series for a general gamma ratio and related properties of the Nørlund–Bernoulli polynomials

D. B. Karp   ; E. G. Prilepkina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 135-158

Résumé

We find an inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable. We give a recurrence relation for the coefficients in terms of the Nørlund–Bernoulli polynomials and determine quite precisely the half-plane of convergence. Our results complement naturally a number of previous investigations of the gamma ratios which began in the 1930ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox's $H$ function in the neighborhood of it's finite singular point. We further apply a particular case of the inverse factorial series expansion to derive a possibly new identity for the Nørlund–Bernoulli polynomials. Bibliography: $49$ titles.

Bibliographie

[1] T. W. Anderson, An introduction to multivariate statistical analysis, Second edition, Wiley, New York, 1984 | MR | Zbl

[2] G. E. P. Box, “A General Distribution Theory for a Class of Likelihood Criteria”, Biometrika, 36 (1949), 317–346 | DOI | MR

[3] B. L. J. Braaksma, “Asymptotic Expansions and Analytic Continuation for a Class of Barnes Integrals”, Composito Math., 15:3 (1962–64), 239–341 | MR

[4] A. Z. Broder, “The $r$-Stirling Numbers”, Discrete Mathematics, 49 (1984), 241–259 | DOI | MR | Zbl

[5] L. Carlitz, “Weighted Stirling numbers of the first and second kind. I”, The Fibonacci Quarterly, 18 (1980), 147–162 | MR | Zbl

[6] L. Carlitz, “Weighted Stirling numbers of the first and second kind. II”, The Fibonacci Quarterly, 18 (1980), 242–257 | MR | Zbl

[7] C. A. Charalambides, Enumerative Combinatorics, Chapman and Hall/CRC, 2002 | MR | Zbl

[8] O. Costin, Asymptotics and Borel summability, Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 141, 2009 | MR | Zbl

[9] A. B. Olde Daalhuis, “Inverse factorial-feries solutions of difference equations”, Proceedings of the Edinburgh Mathematical Society, 47 (2004), 421–448 | DOI | MR | Zbl

[10] E. Delabaere, J.-M. Rasoamanana, “Sommation effective d'une de Borel par séries de factorielles”, Ann. Inst. Fourier (Grenoble), 57:2 (2007), 421–456 | DOI | MR | Zbl

[11] K. Dilcher, “Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials”, J. Approximation Theory, 49 (1987), 321–330 | DOI | MR | Zbl

[12] D. W. H. Gillam, V. P. Gurarii, “On functions uniquely determined by their asymptotic expansion”, Functional Analysis and Its Applications, 40:4 (2006), 273–284 | DOI | MR | Zbl

[13] W. B. Ford, The asymptotic developments of functions defined by Maclaurin series, University of Michigan Studies, Scientific Series, 11, 1936 | Zbl

[14] A. K. Gupta, J. Tang, “On a general distribution for a class of likelihood ratio criteria”, Austraul. J. Statist., 30:3 (1988), 359–366 | DOI | MR | Zbl

[15] H. K. Hughes, “The asymptotic developments of a class of entire functions”, Bull. Amer. Math. Soc., 51 (1945), 456–461 | DOI | MR | Zbl

[16] V. M. Kalinin, “Special Functions and The Limit Properties of Probability Distributions. I”, Investigations in Classical Problems of Probability Theory and Mathematical Statistics, Part I, eds. V. M. Kalinin, O. V. Shalaevskii, Consultants Bureau, New York, 1971, 1–73

[17] D. Karp, E. Prilepkina, “Completely monotonic gamma ratio and infinitely divisible $H$-function of Fox”, Computational Methods and Function Theory, 16 (2016), 135–153 | DOI | MR | Zbl

[18] D. Karp, E. Prilepkina, “Some new facts concerning the delta neutral case of Fox's $H$-function”, Computational Methods and Function Theory, 17:2 (2017), 343–367 | DOI | MR | Zbl

[19] A. A. Kilbas, M. Saigo, $H$-transforms and applications, Analytical Methods and Special Functions, 9, Chapman Hall/CRC, 2004 | DOI | MR | Zbl

[20] M. Koutras, “Non-central stirling numbers and some applications”, Discrete Mathematics, 42 (1982), 73–89 | DOI | MR | Zbl

[21] E. Landau, “Über die Grundlagen der Theorie der Fakultätenreihen”, Sitzsber. Akad. München, 36 (1906), 151–218 | Zbl

[22] L. M. Milne-Thompson, The Calculus of Finite Differences, Macmillan and co. Ltd., 1933 | MR

[23] C. Mitschi, D. Sauzin, Divergent series, monodromy and resurgence, v. I, Lecture Notes in Mathematics, 2153, Springer, 2016 | DOI | MR | Zbl

[24] U. S Nair, “The application of the moment function in the study of distribution laws in statistics”, Biometrika, 30:3/4 (1939), 274–294 | DOI | Zbl

[25] G. Nemes, “Generalization of Binet's gamma function formulas”, Integral Transforms and Special Functions, 24:8 (2013), 597–606 | DOI | MR | Zbl

[26] F. Nevanlinna, “Zur Theorie der Asymptotischen Potenzreihen”, Ann. Acad. Sci. Fenn. Ser. A XII, 3 (1919), 1–81

[27] N. Nielsen, Die Gammafunktion, Chelsea, New York, 1965, originally published by Teubner, Leipzig–Berlin, 1906

[28] N. E. Nørlund, “Sur les séries de facultés”, Acta Math., 37:1 (1914), 327–387 | MR

[29] N. E. Nørlund, Vorlesungen über Differenzrechnung, Springer Verlag, Berlin, 1924

[30] N. E. Nørlund, “Hypergeometric functions”, Acta Mathematica, 94 (1955), 289–349 | DOI | MR

[31] N. E. Nørlund, “sur les valeurs asymptotiques des nombres et des polynômes de Bernoulli”, Rend. Circ. Mat. Palermo, 10:1 (1961), 27–44 | DOI | MR

[32] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010 | MR | Zbl

[33] R. B. Paris, D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001 | MR | Zbl

[34] F. Qi, X-T. Shi, F.-F. Liu, “Expansions of the exponential and the logarithm of power series and applications”, Arab. J. Math., 6:2 (2017), 95–108 | DOI | MR | Zbl

[35] T. D. Riney, “On the coefficients in asymptotic factorial expansions”, Proc. Amer. Math. Soc., 7 (1956), 245–249 | DOI | MR | Zbl

[36] T. D. Riney, “A finite recursion formula for the coefficients in asymptotic expansions”, Trans. Amer. Math. Soc., 88 (1958), 214–226 | DOI | MR | Zbl

[37] T. D. Riney, “Coefficients in certain asymptotic factorial expansions”, Proc. Amer. Math. Soc., 10 (1959), 511–518 | DOI | MR | Zbl

[38] J. Riordan, Combinatorial Identities, John Wiley and Sons, 1968 | MR | Zbl

[39] A. D. Sokal, “An improvement of Watson's theorem on Borel summability”, J. Math. Phys., 21:2 (1980), 261–263 | DOI | MR

[40] F. Tricomi, A. Erdélyi, “The asymptotic expansion of a ratio of gamma functions”, Pacific J. Math., 1:1 (1951), 133–142 | DOI | MR | Zbl

[41] J. G. van der Corput, “On the coefficients in certain asymptotic factorial expansions”, Akad. Wet. (Amsterdam) Proc. Series A, 60:4 (1957), 337–351 | MR | Zbl

[42] H. van Engen, “Concerning gamma function expansions”, Tohoku Math. J., 45 (1939), 124–129

[43] G. N. Watson, “A Theory of Asymptotic Series”, Philosophical Transactions of the Royal Society of London. Series A, 211 (1911), 279–313 | DOI | Zbl

[44] G. N. Watson, “The transformation of an asymptotic series into a convergent series of inverse factorials”, Rend. Giro. Mat. Palermo, 34 (1912), 41–88 | DOI | Zbl

[45] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., 2002 | MR

[46] E. J. Weniger, “Summation of divergent power series by means of factorial series”, Applied Numerical Math., 60 (2010), 1429–1441 | DOI | MR | Zbl

[47] E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function”, J. London Math. Soc., 10 (1935), 287–293 | MR

[48] E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function”, Proc. London Math. Soc. (2), 46 (1940), 389–408 | DOI | MR | Zbl

[49] E. M. Wright, “A recursion formula for the coefficients in an asymptotic expansion”, Glasgow Math. J., 4:1 (1958), 38–41 | DOI | MR | Zbl