Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations

DOI10.1007/s13398-017-0427-2zbMath1439.11088OpenAlexW2743101345MaRDI QIDQ1736959

Dongkyu Lim, Bai-Ni Guo, Feng Qi

Publication date: 26 March 2019

Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s13398-017-0427-2

zbMATH Keywords

derivativegenerating functiondifferential equationStirling numberexplicit formulaFaà di Bruno formulaBell polynomialidentity

Mathematics Subject Classification ID

Bell and Stirling numbers (11B73) Polynomials in number theory (11C08) Special sequences and polynomials (11B83) One-variable calculus (26A06) Exponential and trigonometric functions (33B10)

Related Items (29)

Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions ⋮ The inverse of a triangular matrix and several identities of the Catalan numbers ⋮ Unnamed Item ⋮ Explicit formulas and recurrence relations for generalized Catalan numbers ⋮ Some inequalities and an application of exponential polynomials ⋮ Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials ⋮ Explicit, determinantal, recursive formulas and relations of the Peters polynomials and numbers ⋮ Special values of the Bell polynomials of the second kind for some sequences and functions ⋮ Computation of several Hessenberg determinants ⋮ Causal difference equations with upper and lower solutions in the reverse order ⋮ Some identities related to Eulerian polynomials and involving the Stirling numbers ⋮ Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions ⋮ On multivariate logarithmic polynomials and their properties ⋮ Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers ⋮ Explicit Expressions Related to Degenerate Cauchy Numbers and Their Generating Function ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Some properties of the Hermite polynomials ⋮ Some identities for a sequence of unnamed polynomials connected with the Bell polynomials ⋮ On complete monotonicity for several classes of functions related to ratios of gamma functions ⋮ Generalizations and applications of Young's integral inequality by higher order derivatives ⋮ Unnamed Item ⋮ Some inequalities for generalized Bell-Touchard polynomials ⋮ Unnamed Item ⋮ Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind ⋮ Unnamed Item ⋮ Quasilinear elliptic problem without Ambrosetti–Rabinowitz condition involving a potential in Musielak–Sobolev spaces setting ⋮ Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Mittag--Leffler polynomials ⋮ A new algorithm for finding a common solution of a split variational inequality problem, the fixed point problems and the variational inclusion problems

Cites Work

• Unnamed Item
• Unnamed Item
• Explicit expressions for a family of the Bell polynomials and applications
• Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function
• Trilinear equations, Bell polynomials, and resonant solutions
• Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers
• An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers
• Lah numbers, Laguerre polynomials of order negative one, and the \(n\)th derivative of \(\exp(1/x)\)
• Some inequalities for the Bell numbers
• Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind
• Expansions of the exponential and the logarithm of power series and applications
• Bilinear equations and resonant solutions characterized by Bell polynomials
• Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind
• The Lah Numbers and thenth Derivative of e^1/x
• Six proofs for an identity of the Lah numbers
• Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions
• Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind
• SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS
• Diagonal recurrence relations for the Stirling numbers of the first kind

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