Triple product sums of Catalan triangle numbers

  • Wenchang Chu

Abstract

By means of quadratic transformations for the well-poised hypergeometric series, several reduction and transformation formulae are derived for the triple product sums of Catalan triangle numbers. One of them confirms a conjecture made recently by Miana, Ohtsuka and Romero  [18, 2017].

References

[1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.

[2] X. Chen and W. Chu, Moments on Catalan numbers, J. Math. Anal. Appl. 349:2 (2009), 311–316.

[3] L. Comtet, Advanced Combinatorics, Dordrecht–Holland, The Netherlands, 1974.

[4] R. L. Graham – D. E. Knuth – O. Patashnik, Concrete Mathematics, Addison-Wesley Publ.
Company, Reading, Massachusetts, 1989.

[5] V. J. W. Guo – J. Zeng, Factors of binomial sums from the Catalan triangle, J. Number
Theory 130:1 (2010), 172–186.

[6] J. M. Guti´errez – M. A. Hern´andez – P. J. Miana – N. Romeroa, New identities in the Catalan
triangle, J. Math. Anal Appl. 341:1 (2008), 52–61.

[7] P. Hilton – J. Pedersen, Catalan numbers, their generalization and their uses, Math. Intelligencer
13 (1991), 64–75.

[8] T. Koshy, Catalan Numbers with Applications, Oxford University Press, New York, 2009.

[9] P. J. Miana, H. H. Ohtsuka and N. Romero, Sums of powers of Catalan triangle numbers, Discrete
Mathematics 340 (2017) 2388–2397.

[10] L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), 83–90.

[11] R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Univ. Press, 1999.

[12] A. Slavik, Identities with squares of binomial coefficients, Ars Combinatoria, 113 (2014),
377–383.

[13] Y. Sun and F. Ma, Some new binomial sums related to the Catalan triangle, The Electronic
Journal of Combinatorics 21:1 (2014), #P1.33.

[14] J. Touchard, Sur certains equations functionnelles, Proc. Internat. Math. Congress (Toronto,
1928), Vol.1: 465–472.

[15] F. J. W. Whipple, Some transformations of generalized hypergeometric series, Proc. London
Math. Soc. (Ser.2) 26 (1927), 257–272.
Published
2020-07-30
Section
Articles