Two infinite product formulas with two parameters

Abstract

Two interesting infinite product formulas were presented by Choi et al. [A generalization of Wilf's formula, Kodai Math. J. 26 (2003), pp. 44–48], who generalized an infinite product formula posed by Wilf as a problem in American Mathematical Monthly [H.S. Wilf, Problem 10588, problems and solutions, Amer. Math. Monthly 104 (1997), p. 456]. Here, we aim mainly at establishing two infinite product formulas, expressed in terms of the Gamma function, whose special cases contain the above generalized infinite product formulas. Our formulas presented here are themselves interesting and their another advantage is the production, easily, of various known or (presumably) new formulas as their special cases.

Keywords:

• infinite product formula
• Euler–Mascheroni constant
• Gamma function
• Psi function
• asymptotic formulas

2010 Mathematics Subject Classification :

• Primary: 33B15
• Secondary: 33C05

Acknowledgements

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).