Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds

Feng Qi, Da-Wei Niu, and Dongkyu Lim, Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds, Miskolc Mathematical Notes 20 (2019), no. 2, 1129--1137; available online at https://doi.org/10.18514/MMN.2019.2976.International audienceIn 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

Closed formulas and identities on the Bell polynomials and falling factorials

The authors establish a pair of closed-form expressions for special values of the Bell polynomials of the second kind for the falling factorials, derive two pairs of identities involving the falling factorials, find an equivalent expression between two special values for the Bell polynomials of the second kind, and present five closed-form expressions for the (modified) spherical Bessel functions

SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND FOR SOME SEQUENCES AND FUNCTIONS: Special values of Bell polynomials of second kind

In the paper, the authors concisely review some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, explicitly present closed formulas for those sequences investigated in [F. T. Howard, A special class of Bell polynomials, Math. Comp. 35 (1980), no. 151, 977–989; Available online at https://doi.org/10.2307/2006208], and newly establish some closed formulas for some special values of the Bell polynomials of the second kind