Taylor expansions for the generating function of Catalan-like numbers

Lily Li Liu,Xiaoli Li,Hari M Srivastava

doi:10.1080/23311835.2016.1200305

Abstract

In 2002, Eu, Liu and Yeh introduced new Taylor expansions of the generating function of Catalan and Motzkin numbers. And they presented that this Taylor style expansion can be applied to more generating functions satisfying some relations (Advances in Applied Mathematics, 29 (2002) 345–357). In this paper, we focus on this Taylor expansion of the generating function of Catalan-like numbers, which are common generalizations of many classic counting coefficients, such as the Catalan numbers, the Motzkin numbers and the Schroder numbers. We present the recurrence relations of the coefficients and bivariate generating functions of the remainders of the new Taylor expansion of the generating function of Catalan-like numbers.

Highlights

The Taylor expansion of a real function f(x), which is infinitely differential, at the real number 0 is f (x) = n−1 ∑ i=0 f (i)(0) i! xi + Rn(x), (1)(wnh−er1e)Rthn(Txa)yislotrhpeonlythnoremmiaali∑ndnie=−r01, which is x f (i)(0) i. i!the error incurred in approximating the function f(x) by its Traditionally, the remainders of Taylor expansions play a central role in the theory of functions, numberical approximations, asymptotic expansions, etc
Liu and Yeh introduced new Taylor expansions, whose remainders involve the function itself, of the generating function of Catalan and Motzkin numbers. We focus on this Taylor expansion of the generating function of Catalanlike numbers, which are the 0th column of special cases of Riordan arrays
We present the recurrence relations of the coefficients and bivariate generating functions of the remainders of the new Taylor expansion of the generating function of Catalan-like numbers

Summary

Introduction

Liu and Yeh introduced new Taylor expansions, whose remainders involve the function itself, of the generating function of Catalan and Motzkin numbers. We focus on this Taylor expansion of the generating function of Catalanlike numbers, which are the 0th column of special cases of Riordan arrays. We present the recurrence relations of the coefficients and bivariate generating functions of the remainders of the new Taylor expansion of the generating function of Catalan-like numbers. This new Taylor expansions can be generalized to the Catalan-like numbers (Eu et al, 2002).

Results

Conclusion