An integral representation of the Gauss hypergeometric functions and its applications

References

[1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopaedia Math. Sci. 71, Cambridge University, Cambridge, 1999. 10.1017/CBO9781107325937Search in Google Scholar

[2] L.-Y. Bao, D. Lim and F. Qi, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, submitted. Search in Google Scholar

[3] Y. A. Brychkov, Handbook of Special Functions, CRC Press, Boca Raton, 2008. 10.1201/9781584889571Search in Google Scholar

[4] C. A. Charalambides, Enumerative Combinatorics, CRC Press Ser. Discrete Math. Appl., Chapman & Hall/CRC, Boca Raton, 2002. Search in Google Scholar

[5] I. Chiţescu, Around the Formula of Faà di Bruno, Éditions Universitaires Européennes, Mauritius, 2017. Search in Google Scholar

[6] L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1974. 10.1007/978-94-010-2196-8Search in Google Scholar

[7] K. A. Driver and S. J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006), 115–120. Search in Google Scholar

[8] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. I, Robert E. Krieger, Melbourne, 1981. Search in Google Scholar

[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015. Search in Google Scholar

[10] Y.-W. Li and F. Qi, A new closed-form formula of the Gauss hypergeometric function at specific arguments, Axioms 13 (2024), no. 5, Paper No. 317. 10.3390/axioms13050317Search in Google Scholar

[11] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren Math. Wiss. 52, Springer, New York, 1966. 10.1007/978-3-662-11761-3Search in Google Scholar

[12] G. V. Milovanović and F. Qi, Closed-form formulas of two Gauss hypergeometric functions of specific parameters, J. Math. Anal. Appl. 543 (2025), no. 2, Paper No. 129024. 10.1016/j.jmaa.2024.129024Search in Google Scholar

[13] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University, Cambridge, 2010. Search in Google Scholar

[14] J. W. Pearson, Computation of hypergeometric functions, Dissertation, University of Oxford, 2009. Search in Google Scholar

[15] F. Qi, Power series expansion of Wilf function, preprint (2021), https://arxiv.org/abs/2110.08576. Search in Google Scholar

[16] F. Qi and B.-N. Guo, Explicit and recursive formulas, integral representations, and properties of the large Schröder numbers, Kragujevac J. Math. 41 (2017), no. 1, 121–141. 10.5937/KgJMath1701121FSearch in Google Scholar

[17] F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Paper No. 140. 10.1007/s00009-017-0939-1Search in Google Scholar

[18] F. Qi, D.-W. Niu, D. Lim and Y.-H. Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, J. Math. Anal. Appl. 491 (2020), no. 2, Article ID 124382. 10.1016/j.jmaa.2020.124382Search in Google Scholar

[19] F. Qi, X.-T. Shi and B.-N. Guo, Two explicit formulas of the Schröder numbers, Integers 16 (2016), Paper No. A23. 10.1016/j.ajmsc.2016.06.002Search in Google Scholar

[20] F. Qi and P. Taylor, Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials, Appl. Anal. Discrete Math. 18 (2024), no. 1, 92–115. 10.2298/AADM230902020QSearch in Google Scholar

[21] M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series F 1 2 and F 2 3 with applications, Integral Transforms Spec. Funct. 22 (2011), no. 11, 823–840. 10.1080/10652469.2010.549487Search in Google Scholar

[22] R. Sivaraman, M. MuniruIddrisu and J. López-Bonilla, On an expression for F 1 2 ⁢ ( n + 1 2 , n + 1 ; n + 3 2 ; - 1 ) , Int. J. Appl. Adv. Sci. Res. 9 (2024), no. 2, 113–114. Search in Google Scholar

[23] N. M. Temme, Special Functions, John Wiley & Sons, New York, 1996. 10.1002/9781118032572Search in Google Scholar

[24] M. D. Ward, Asymptotic rational approximation to pi: Solution of an “unsolved problem” posed by Herbert Wilf, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, Discrete Math. Theor. Comput. Sci. Proc., Association Discrete Mathematics and Theoretical Computer Science, Nancy (2010), 591–601. 10.46298/dmtcs.2777Search in Google Scholar

[25] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis—An Introduction to the General Theory of Infinite Processes and of Analytic Functions with an Account of the Principal Transcendental Functions, 5th ed., Cambridge University, Cambridge, 2021. Search in Google Scholar