Abstract
In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, conclude the inverse of an integer, unit, and lower triangular matrix, derive an inversion theorem, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials respectively.
Highlights
In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, conclude the inverse of an integer, unit, and lower triangular matrix, derive an inversion theorem, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials respectively
One calls 2F1(a, b; c; z) the classical hypergeometric function. It is well known [7, 39, 47] that the Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as
The Faa di Bruno formula can be described in terms of the Bell polynomials of the second kind Bn,k(x1, x2, . . . , xn−k+1) by dn dtn f ◦ h(t) =
Summary
It is common knowledge [5, 11, 42] that the generalized hypergeometric series p Fq We will establish two identities to express the generating function F (t) of the Chebyshev polynomials of the second kind Uk(x) and its higher order derivatives F (k)(t) in terms of F (k)(t) and F (t) each other, find an explicit formula and an identity for the Chebyshev polynomials of the second kind Uk(x), derive the inverse of an integer, unit, and lower triangular matrix, acquire an inversion theorem, present several identities of the Catalan numbers Ck, and give some remarks on the closely related results including connections of the Catalan numbers Ck with the Chebyshev polynomials of the second kind Uk(x), the central Delannoy numbers [20, 21], and the Fibonacci polynomials [20, 24] respectively
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