Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 074, 13 pages      arXiv:2403.14942      https://doi.org/10.3842/SIGMA.2024.074
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

Asymptotics of the Humbert Function $\Psi_1$ for Two Large Arguments

Peng-Cheng Hang and Min-Jie Luo
Department of Mathematics, School of Mathematics and Statistics, Donghua University, Shanghai 201620, P.R. China

Received March 27, 2024, in final form August 02, 2024; Published online August 09, 2024

Abstract
Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions $\Phi_2$, $\Phi_3$ and $\Xi_2$ for two large arguments, but their technique cannot handle the Humbert function $\Psi_1$. In this paper, we establish the leading asymptotic behavior of the Humbert function $\Psi_1$ for two large arguments. Our proof is based on a connection formula of the Gauss hypergeometric function and Nagel's approach (2004). This approach is also applied to deduce asymptotic expansions of the generalized hypergeometric function $_pF_q$ $(p\leqslant q)$ for large parameters, which are not contained in NIST handbook.

Key words: Humbert function; asymptotics; generalized hypergeometric function.

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References

  1. Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions with formulas, graphs, and mathematical tables, NBS Appl. Math. Ser., Vol. 55, U.S. Government Printing Office, Washington, DC, 1964.
  2. Ahbli K., Mouayn Z., A generating function and formulae defining the first-associated Meixner-Pollaczek polynomials, Integral Transforms Spec. Funct. 29 (2018), 352-366, arXiv:1708.03358.
  3. Belafhal A., Saad F., Conversion of circular beams by a spiral phase plate: generation of generalized Humbert beams, Optik 138 (2017), 516-528.
  4. Borghi R., ''Analytical continuation'' of flattened Gaussian beams, J. Opt. Soc. Amer. A 40 (2023), 816-823.
  5. Brychkov relaxYu.A., Saad N., On some formulas for the Appell function $F_2(a,b,b';c,c';w;z)$, Integral Transforms Spec. Funct. 25 (2014), 111-123.
  6. Bühring W., The behavior at unit argument of the hypergeometric function $_3F_2$, SIAM J. Math. Anal. 18 (1987), 1227-1234.
  7. Bühring W., Generalized hypergeometric functions at unit argument, Proc. Amer. Math. Soc. 114 (1992), 145-153.
  8. Butzer P.L., Kilbas A.A., Trujillo J.J., Stirling functions of the second kind in the setting of difference and fractional calculus, Numer. Funct. Anal. Optim. 24 (2003), 673-711.
  9. Carlson B.C., Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 17 (1966), 32-39.
  10. Choi J., Hasanov A., Applications of the operator $H(\alpha,\beta)$ to the Humbert double hypergeometric functions, Comput. Math. Appl. 61 (2011), 663-671, arXiv:0810.3796.
  11. El Halba E.M., Nebdi H., Boustimi M., Belafhal A., On the Humbert confluent hypergeometric function used in laser field, Phys. Chem. News 73 (2014), 90-93.
  12. Fields J.L., Confluent expansions, Math. Comp. 21 (1967), 189-197.
  13. Hang P.-C., Luo M.-J., Asymptotics of Saran's hypergeometric function $F_K$, J. Math. Anal. Appl. 541 (2025), 128707, 19 pages, arXiv:2405.00325.
  14. Humbert P., IX.-The confluent hypergeometric functions of two variables, Proc. Roy. Soc. Edinburgh 41 (1922), 73-96.
  15. Joshi C.M., Arya J.P., Inequalities for certain hypergeometric functions, Math. Comp. 38 (1982), 201-205.
  16. Jurš.enas R., On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series, Lith. Math. J. 54 (2014), 61-73, arXiv:1301.3039.
  17. Knottnerus U.J., Approximation formulae for generalized hypergeometric functions for large values of the parameters, J. B. Wolters, Groningen, 1960.
  18. Lin Y., Wong R., Asymptotics of generalized hypergeometric functions, in Frontiers in Orthogonal Polynomials and $q$-Series, Contemp. Math. Appl. Monogr. Expo. Lect. Notes, Vol. 1, World Scientific Publishing, Hackensack, NJ, 2018, 497-521.
  19. Luke Y.L., The special functions and their approximations. Vol. II, Math. Sci. Eng., Vol. 53, Academic Press, New York, 1969.
  20. Luo M.-J., Raina R.K., On certain results related to the hypergeometric function $F_K$, J. Math. Anal. Appl. 504 (2021), 125439, 18 pages.
  21. Nagel B., Confluence expansions of the generalized hypergeometric function, J. Math. Phys. 45 (2004), 495-508.
  22. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A. (Editors), NIST digital library of mathematical functions, Release 1.2.1 of 2024-06-15, aviable at https://dlmf.nist.gov/.
  23. Paris R.B., A Kummer-type transformation for a $_2F_2$ hypergeometric function, J. Comput. Appl. Math. 173 (2005), 379-382.
  24. Pólya G., SzegHo G., Problems and theorems in analysis. Vol. II: Theory of functions, zeros, polynomials, determinants, number theory, geometry, Classics Math., Springer, Berlin, 1998.
  25. Van Gorder R.A., Computation of certain infinite series of the form $\sum f(n)n^k$ for arbitrary real-valued $k$, Appl. Math. Comput. 215 (2009), 1209-1216.
  26. Wald S., Henkel M., On integral representations and asymptotics of some hypergeometric functions in two variables, Integral Transforms Spec. Funct. 29 (2018), 95-112, arXiv:1707.06275.

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