On several new closed-form evaluations for the generalized hypergeometric functions

Document Type : Original paper

Authors

1 Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education

2 Department of Mathematics Education, Andong National University

3 Vedant College of Engineering and Technology (Rajasthan Technical University)

Abstract

The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function $_{q+1}F_q(z)$ for $q= 2, 3$. This is achieved by means of separating the generalized hypergeometric function $_{q+1}F_q(z)$ for $q=1, 2, 3$ into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji.

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References

[1] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge; Reprinted by Stechert-Hafner, New York, (1964), 1935.
[2] H. Exton, Multiple Hypergeometric Functions, Ellis Horwood, Chichester, UK, 1976.
[3] Andrews G.E., R. Askey, and R. Roy, Special Functions, Oxford University Press, Cambridge, 2000.
[4] H.W. Gould, Combinatorial Identities, Published by the author, Revised edition, 1972.
[5] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Cambridge, 2008.
[6] B.R. Kumar, A. Kılı¸cman, and A.K. Rathie, Applications of Lehmer’s infinite series involving reciprocals of the central binomial coefficients, J. Function Spaces 2022 (2022), ID: 1408543.
[7] D.H. Lehmer, Interesting series involving the central binomial coefficient, The Amer. Math. Monthly 92 (1985), no. 7, 449–457.
[8] T. Mansour, Combinatorial identities and inverse binomial coefficients, Adv. Appl. Math. 28 (2002), no. 2, 196–202.
[9] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, Handbook of Mathematical Functions, Oxford University Press, Cambridge, 2010.
[10] J. Pla, The sum of inverses of binomial coefficients revisited, Fibonacci Quart. 35 (1997), no. 4, 342–345.
[11] A.P. Prudnikov, Y.A. Brychkov, and O.I. Marichev, More Special Functions (Integrals and Series vol 3), 1990.
[12] E.D. Raniville, Special Functions, The Macmillan Company, New York; Reprinted by Chelsea Publishing Company, Bronx, NY, (1971), 1960.
[13] J. Riordan, Combinatorial Identities, Wiley, 1968.
[14] T. Sherman, Summations of Glaisher’s and Apery Like Series, Available at http://math.arizona.edu/ura/001/sherman.travis/series.pdf, 2000.
[15] L.J. Slater, Generalized Hypergeometric Functions, Oxford University Press, Cambridge, 1960.
[16] R. Sprugnoli, Combinatorial Identities, http://www.dsi.unifi.it/ resp/GouldBK. pdf.
[17] R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: Electronic Journal of Combinatorial Number Theory 6 (2006), #A27.
[18] B. Sury, Sum of the reciprocals of the binomial coefficients, European J. Combin. 14 (1993), no. 4, 351–353.
[19] B. Sury, T. Wang, and F.Z. Zhao, Identities involving reciprocals of binomial coefficients, J. Integer Seq. 7 (2004), no. 2, ID: 04.2.8.
[20] T. Trif, Combinatorial sums and series involving inverses of binomial coefficients, Fibonacci Quart. 38 (2000), no. 1, 79–84.
[21] A.D. Wheelon, On the summation of infinite series in closed form, J. Appl. Physics 25 (1954), no. 1, 113–118.
[22] L. Zhang and W. Ji, The series of reciprocals of non-central binomial coefficients, Amer. J. Comput. Math. 3 (2013), no. 3, 31–37.
[23] F.Z. Zhao and T. Wang, Some results for sums of the inverses of binomial coefficients, Integers: Electronic Journal of Combinatorial Number Theory 5 (2005), no. 1, #A22.
 
Communications in Combinatorics and Optimization
Volume 8, Issue 4
December 2023
Pages 737-749

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