Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials. 1. Preliminaries It is common knowledge [8, 15, 52] that the generalized hypergeometric series pFq(a1, . . . , ap; b1, . . . , bq; z) = ∞ ∑ n=0 (a1)n · · · (ap)n (b1)n · · · (bq)n z n! is defined for complex numbers ai ∈ C and bi ∈ C \ {0,−1,−2, . . . }, for positive integers p, q ∈ N, and in terms of the rising factorials (x)n defined by (x)n = n−1 ∏ `=0 (x+ `) = { x(x+ 1) · · · (x+ n− 1), n ≥ 1; 1, n = 0. Specially, one calls 2F1(a, b; c; z) the classical hypergeometric function. It is well known [12, 47, 55] that the Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n−2 triangles if different orientations are counted separately? whose solution is the Catalan number Cn−2”. The Catalan numbers Cn can be generated by 2 1 + √ 1− 4x = 1− √ 1− 4x 2x = ∞ ∑ n=0 Cnx n = 1 + x+ 2x + 5x + · · · and explicitly expressed as Cn = 1 n+ 1 ( 2n n ) = 2F1(1− n,−n; 2; 1) = 4Γ(n+ 1/2) √ π Γ(n+ 2) , 2010 Mathematics Subject Classification. Primary 11B83; Secondary 05A15, 05A19, 11C08, 11C20, 11Y35, 15A09, 15B36, 33C05, 34A34.