Transformation formulae for terminating balanced $_4F_3$-series and implications

Abstract

A new transformation from terminating balanced $_4F_3$-series to $_3F_2$-series is proved that contains a few known summation formulae as special cases. By means of Whipple's transformation, further closed form evaluations are given for terminating well--poised $_7F_6$-series as applications.

Keywords

• Classical hypergeometric series
• balanced series
• well–poised series
• Whipple’s formula
• Pfaff–Saalschütz formula

References

1. [1] W.N. Bailey, Some identities involving generalized hypergeometric series, Proc. London Math. Soc. 29, 503–516, 1929.
2. [2] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
3. [3] Yury A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2008.
4. [4] L. Carlitz, Summation of a special $_4F_3$, Boll. Union Mat. Ital. 18, 90–93, 1963.
5. [5] W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283, 31–57, 1994.
6. [6] W. Chu, Inversion techniques and combinatorial identities: a unified treatment for the $_7F_6$-series identities, Collect. Math. 45 (1), 13–43, 1994.
7. [7] W. Chu, Binomial convolutions and hypergeometric identities, Rend. Circolo Mat. Palermo (serie 2) 43, 333–360, 1994.
8. [8] W. Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J. Math. 32 (2), 561–587, 2002.

9. [9] W. Chu, Analytical formulae for extended $_3F_2$-series of Watson–Whipple–Dixon with two extra integer parameters, Math. Comp. 81 (277), 467–479, 2012.
10. [10] W. Chu, Terminating $_4F_3$-series extended with two integer parameters, Integral Transforms Spec. Funct. 27 (10), 794–805, 2016.
11. [11] J. Dougall, On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25, 114–132, 1907.
12. [12] I.M. Gessel, Finding identities with the WZ method, J. Symbolic Comput. 20 (5/6), 537–566, 1995.
13. [13] I.M. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (2), 295–308, 1982.
14. [14] A. Ishkhanyan and C. Cesarano, Generalized-hypergeometric solutions of the general Fuchsian linear ODE having five regular singularities, Axioms 8, (102), 2019.
15. [15] A. Lupica, C. Cesarano, F. Crisanti, and A. Ishkhanyan, Analytical solution of the three-dimensional Laplace equation in terms of linear combinations of hypergeometric functions, Mathematics 9, (3316), 2021.
16. [16] I.D. Mishev, Relations for a class of terminating $_4F_3(4)$ hypergeometric series, Integral Transforms Spec. Funct. 33 (3), 199–215, 2022.
17. [17] C.Y. Wang and X.J. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
18. [18] F.J.W. Whipple, A group of generalized hypergeometric series: Relations between 120 allied series of type $F[a,b,c;d,e]$, Proc. London Math. Soc. (Ser.2) 23, 104–114, 1925.
19. [19] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (Ser.2) 24, 247–263, 1926.
20. [20] D. Zeilberger, Forty “strange" computer–discovered and computer–proved (of course) hypergeometric series evaluations, Available at http://www.math.rutgers.edu/ ~zeilberg/ekhad/ekhad.html.