Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds
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In this article, the authors derived two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomial of the second kind, making use of a new inversion theorem for combinatorial coefficients.Abstract:
In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.read more
Citations
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Special values of the bell polynomials of the second kind for some sequences and functions
TL;DR: In this article, after concisely surveying some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, the authors newly established some new closed formulas for some specific values of Bell polynomials.
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Simplifying coefficients in differential equations for generating function of Catalan numbers
Feng Qi,Yonghong Yao +1 more
TL;DR: In this paper, by the Faa di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients in two families of nonlinear ODEs.
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Alternative proofs of some formulas for two tridiagonal determinants
TL;DR: In this article, Qi et al. provided five alternative proofs of two formulas for a tridiagonal determinant, and provided a detailed proof of the inverse of the corresponding tridagonal matrix.
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Several Explicit and Recurrent Formulas for Determinants of Tridiagonal Matrices via Generalized Continued Fractions
TL;DR: In this paper, the authors present several explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these newly established formulas to evaluations for the Sylvester matrix and two sylvester type matrices.
References
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Handbook of Mathematical Functions
Book
Advanced Combinatorics: The Art of Finite and Infinite Expansions
Louis Comtet,J. W. Nienhuys +1 more
TL;DR: A vocabulary of combinatorial analysis can be found in this paper, where the authors define definitions of partitions of an integer [n]- 22 Generating Functions of p(n) and P(n, m)- 23 Conditional Partitions- 24 Ferrers Diagrams- 25 Special Identities 'Formal' and 'Combinatorial' Proofs- 26 Partitions with Forbidden Summands Denumerants- Supplement and Exercises- III Identities and Expansions- III Identity and Expansion of a Product of Sums Abel Identity- 31
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CRC Standard Mathematical Tables and Formulae
TL;DR: In this article, the authors present an algebraic version of elementary mathematics proofs without words, including the concept of special numbers and the notion of numbers without words in elementary algebra.
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CRC standard mathematical tables and formulae
TL;DR: In this article, the authors present a taxonomy of conversion factors for probability and statistics financial tables, including conversion factors and conversion factors in the context of taxonomic analysis of financial transactions.
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