scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Sine Function ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Sine Functions

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Cosine Function ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Hyperbolic Cosine Function

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius-Euler Numbers

2018 ◽  
Vol 72 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Euler Numbers ◽  
Inversion Formulas ◽  
Faà Di Bruno Formula

Abstract In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

scholarly journals Three closed forms for convolved Fibonacci numbers

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Generating Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faa di Bruno formula and several properties of the Bell polynomials of the second kind, the author computes higher order derivatives of the generating function of convolved Fibonacci numbers and, consequently, derives three closed forms for convolved Fibonacci numbers in terms of the falling and rising factorials, the Lah numbers, the signed Stirling numbers of the first kind, and the golden ratio.

scholarly journals Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

10.37236/2476 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Combinatorial Identities ◽  
Convolution Type ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Inverse Relation

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

scholarly journals Simplifying coefficients in a family of nonlinear ordinary differential equations

2019 ◽  
Vol 22 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Combinatorial Analysis ◽  
Bell Polynomials ◽  
Nonlinear Ordinary Differential Equations ◽  
Faà Di Bruno Formula

By virtue of the Faá di Bruno formula, properties of the Stirling numbers and the Bell polynomials of the second kind, the binomial inversion formula, and other techniques in combinatorial analysis, the author finds a simple, meaningful, and signicant expression for coefficients in a family of nonlinear ordinary differential equations.

scholarly journals Simple forms for coefficients in two families of ordinary differential equations

2018 ◽  
Vol 6 (1) ◽  
pp. 7
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Faà Di Bruno Formula

In the paper, by virtue of the Faá di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion formula for the Stirling numbers of the first and second kinds, the author finds simple, meaningful, and significant forms for coefficients in two families of ordinary differential equations.

scholarly journals On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities

2017 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Recurrence Relations ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Logarithmic Convexity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Faà Di Bruno Formula ◽  
Determinantal Inequalities

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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