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. In this paper, we obtain new generating functions involvingfamilies of pairs of inverse functions by using a generalization of the Sri-vastava’s theorem [H. M. Srivastava, Some generalizations of Carlitz’stheorem, Pacic J. Math. 85 (1979), 471–477] obtained by Tremblay andFug`ere [Generating functions related to pairs of inverse functions, Trans-form methods and special functions, Varna ’96, Bulgarian Acad. Sci.,Soa (1998), 484–495]. Special cases are given. These can be seen as gen-eralizations of the generalized Bernoulli polynomials and the generalizeddegenerate Bernoulli polynomials. 1. IntroductionIn 1977, motivated by the work of Srivastava and Singhal [27] on the Ja-cobi polynomials, Carlitz obtained the following generating function for theLaguerre polynomials [2, p. 525, Eq.(5.5)]:X ∞n=0 L (α+λn)n (x +ny)t n =(1+ω) α+1 e −xω 1−λω +ω(1+ω)y(1.1) ,where α,λ are arbitrary complex numbers and ω is a function of t dened by(1.2) ω = t(1+ω) λ+1 e −yωwith ω(0) = 0.In the same paper, Carlitz [2, p. 521, Theorem 1 and Eq.(2.10)] extendedthese results to the formsX

daehan suhak hoebo/bulletin of the korean mathematical society

A FURTHER INVESTIGATION OF GENERATING FUNCTIONS RELATED TO PAIRS OF INVERSE FUNCTIONS WITH APPLICATIONS TO GENERALIZED DEGENERATE BERNOULLI POLYNOMIALS