The inverse of a triangular matrix and several identities of the Catalan numbers. (English) Zbl 1513.11091
Summary: 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.
MSC:
| 11B83 | Special sequences and polynomials |
| 05A15 | Exact enumeration problems, generating functions |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
Keywords:
Chebyshev polynomials of the second kind; inverse matrix; Catalan number; generating function; integral representationReferences:
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