Computers and Mathematics with Applications, ISSN 0898-1221, 2006, Volume 51, Issue 3, pp. 631 - 642

Recently, Srivastava and Pintér [1] investigated several interesting properties and relationships involving the classical as well as the generalized (or...

Euler polynomials and numbers | ernoulli polynomials and numbers | Generalized (or higher-order) Euler polynomials and numbers, Apostol-Bernoulli polynomials and numbers, Apostol-Euler polynomials and numbers, Generalized Apostol-Bernoulli polynomials and numbers, Generalized Apostol-Euler polynomials and numbers, Stirling numbers of the second kind, Generating functions, Srivastava-Pintér addition theorems, Recursion formulas | Generalized (or higher-order) Bernoulli polynomials and numbers | Generalized (or higher-order) Euler polynomials and numbers, Apostol-Bernoulli polynomials and numbers, Apostol-Euler polynomials and numbers, Generalized Apostol-Bernoulli polynomials and numbers, Generalized Apostol-Euler polynomials and numbers | Bernoulli polynomials and numbers | MATHEMATICS, APPLIED | generalized (or higher-order) Euler polynomials and numbers | stirling numbers of the second kind | generalized Apostol-Bernoulli polynomials and numbers | Srivastava-Pinter addition theorems | generating functions | generalized (or higher-order) Bernoulli polynomials and numbers | generalized Apostol-Euler polynomials and numbers | recursion formulas | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Apostol-Euler polynomials and numbers | Apostol-Bernoulli polynomials and numbers | Mathematical models

Euler polynomials and numbers | ernoulli polynomials and numbers | Generalized (or higher-order) Euler polynomials and numbers, Apostol-Bernoulli polynomials and numbers, Apostol-Euler polynomials and numbers, Generalized Apostol-Bernoulli polynomials and numbers, Generalized Apostol-Euler polynomials and numbers, Stirling numbers of the second kind, Generating functions, Srivastava-Pintér addition theorems, Recursion formulas | Generalized (or higher-order) Bernoulli polynomials and numbers | Generalized (or higher-order) Euler polynomials and numbers, Apostol-Bernoulli polynomials and numbers, Apostol-Euler polynomials and numbers, Generalized Apostol-Bernoulli polynomials and numbers, Generalized Apostol-Euler polynomials and numbers | Bernoulli polynomials and numbers | MATHEMATICS, APPLIED | generalized (or higher-order) Euler polynomials and numbers | stirling numbers of the second kind | generalized Apostol-Bernoulli polynomials and numbers | Srivastava-Pinter addition theorems | generating functions | generalized (or higher-order) Bernoulli polynomials and numbers | generalized Apostol-Euler polynomials and numbers | recursion formulas | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Apostol-Euler polynomials and numbers | Apostol-Bernoulli polynomials and numbers | Mathematical models

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 05/2017, Volume 40, Issue 7, pp. 2347 - 2361

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial...

functional equations | central factorial numbers | binomial coefficients | array polynomials | Euler numbers and polynomials | generating functions | binomial sum | Stirling numbers | combinatorial sum | MATHEMATICS, APPLIED | IDENTITIES | ARRAY TYPE POLYNOMIALS | GENERATING-FUNCTIONS | BERNOULLI | Computation | Factorials | Mathematical analysis | Mathematical models | Polynomials | Arrays | Combinatorial analysis | Sums | Mathematics - Number Theory

functional equations | central factorial numbers | binomial coefficients | array polynomials | Euler numbers and polynomials | generating functions | binomial sum | Stirling numbers | combinatorial sum | MATHEMATICS, APPLIED | IDENTITIES | ARRAY TYPE POLYNOMIALS | GENERATING-FUNCTIONS | BERNOULLI | Computation | Factorials | Mathematical analysis | Mathematical models | Polynomials | Arrays | Combinatorial analysis | Sums | Mathematics - Number Theory

Journal Article

Aequationes mathematicae, ISSN 0001-9054, 12/2017, Volume 91, Issue 6, pp. 1055 - 1071

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated...

Combinatorial identities | Generating function | Mathematics | Secondary 11B68 | Primary 11B83 | Analysis | Poly-Bernoulli numbers | 11B73 | 05A19 | Combinatorics | Incomplete r -Stirling numbers | 05A15 | Poly-Cauchy numbers | Incomplete r-Stirling numbers | POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | BESSEL NUMBERS | COMBINATORICS

Combinatorial identities | Generating function | Mathematics | Secondary 11B68 | Primary 11B83 | Analysis | Poly-Bernoulli numbers | 11B73 | 05A19 | Combinatorics | Incomplete r -Stirling numbers | 05A15 | Poly-Cauchy numbers | Incomplete r-Stirling numbers | POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | BESSEL NUMBERS | COMBINATORICS

Journal Article

Journal of Number Theory, ISSN 0022-314X, 11/2016, Volume 168, pp. 117 - 127

The Delannoy numbers and Schröder numbers are given byDn=∑k=0n(nk)(n+kk)andSn=∑k=0n(nk)(n+kk)1k+1, respectively. Let p>3 be a prime. We mainly prove...

Delannoy numbers | Schröder numbers | Bernoulli numbers | Supercongruence | POLYNOMIALS | MATHEMATICS | Schroder numbers | CONGRUENCES

Delannoy numbers | Schröder numbers | Bernoulli numbers | Supercongruence | POLYNOMIALS | MATHEMATICS | Schroder numbers | CONGRUENCES

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2014, Volume 244, pp. 149 - 157

The main objective in this paper is first to establish new identities for the λ-Stirling type numbers of the second kind, the λ-array type polynomials, the...

Bernoulli polynomials and Bernoulli numbers | Apostol–Bernoulli polynomials and Apostol–Bernoulli numbers | [formula omitted]-Stirling numbers of the second kind | [formula omitted]-Array polynomials | [formula omitted]-Bell numbers and [formula omitted]-Bell polynomials | λ-Bell numbers and λ-Bell polynomials | λ-Array polynomials | λ-Stirling numbers of the second kind | Apostol-Bernoulli polynomials and Apostol-Bernoulli numbers | MATHEMATICS, APPLIED | lambda-Array polynomials | lambda-Stirling numbers of the second kind | lambda-Bell numbers and lambda-Bell polynomials | APOSTOL-BERNOULLI | GENERATING-FUNCTIONS | EULER

Bernoulli polynomials and Bernoulli numbers | Apostol–Bernoulli polynomials and Apostol–Bernoulli numbers | [formula omitted]-Stirling numbers of the second kind | [formula omitted]-Array polynomials | [formula omitted]-Bell numbers and [formula omitted]-Bell polynomials | λ-Bell numbers and λ-Bell polynomials | λ-Array polynomials | λ-Stirling numbers of the second kind | Apostol-Bernoulli polynomials and Apostol-Bernoulli numbers | MATHEMATICS, APPLIED | lambda-Array polynomials | lambda-Stirling numbers of the second kind | lambda-Bell numbers and lambda-Bell polynomials | APOSTOL-BERNOULLI | GENERATING-FUNCTIONS | EULER

Journal Article

Publicationes Mathematicae, ISSN 0033-3883, 2016, Volume 88, Issue 3-4, pp. 357 - 368

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their...

Restricted Stirling numbers | Restricted poly-bernoulli numbers | Bernoulli numbers | Associated poly-bernoulli numbers | Poly-Bernoulli numbers | Stirling numbers | Associated stirling numbers | POLYNOMIALS | MATHEMATICS | associated Stirling numbers | associated poly-Bernoulli numbers | CAUCHY NUMBERS | restricted Stirling numbers | poly-Bernoulli numbers | restricted poly-Bernoulli numbers

Restricted Stirling numbers | Restricted poly-bernoulli numbers | Bernoulli numbers | Associated poly-bernoulli numbers | Poly-Bernoulli numbers | Stirling numbers | Associated stirling numbers | POLYNOMIALS | MATHEMATICS | associated Stirling numbers | associated poly-Bernoulli numbers | CAUCHY NUMBERS | restricted Stirling numbers | poly-Bernoulli numbers | restricted poly-Bernoulli numbers

Journal Article

Applicable Analysis and Discrete Mathematics, ISSN 1452-8630, 4/2018, Volume 12, Issue 1, pp. 1 - 35

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known...

Integers | Numbers | Generating function | Factorials | Discrete mathematics | Polynomials | Coefficients | Combinatorics | New family | Combinatorial sum | Central factorial numbers | Bernoulli numbers | Binomial coefficients | Euler numbers | Functional equations | Generating functions | Array polynomials | Stirling numbers | Fibonacci numbers | MATHEMATICS, APPLIED | COMBINATORIAL SUMS | Q-BERNOULLI NUMBERS | GENERATING-FUNCTIONS | MATHEMATICS

Integers | Numbers | Generating function | Factorials | Discrete mathematics | Polynomials | Coefficients | Combinatorics | New family | Combinatorial sum | Central factorial numbers | Bernoulli numbers | Binomial coefficients | Euler numbers | Functional equations | Generating functions | Array polynomials | Stirling numbers | Fibonacci numbers | MATHEMATICS, APPLIED | COMBINATORIAL SUMS | Q-BERNOULLI NUMBERS | GENERATING-FUNCTIONS | MATHEMATICS

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2015, Volume 268, pp. 844 - 858

In the paper, by induction, the Faà di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher...

Bell polynomial of the second kind | Tangent number | Bernoulli number | Explicit formula | Derivative polynomial | Euler polynomial | MATHEMATICS, APPLIED | BERNOULLI NUMBERS | INEQUALITIES | IDENTITIES | COMPLETE MONOTONICITY | STIRLING NUMBERS | POLYNOMIALS | EXPLICIT FORMULAS | INTEGRAL-REPRESENTATION | 2ND KIND | 1ST KIND

Bell polynomial of the second kind | Tangent number | Bernoulli number | Explicit formula | Derivative polynomial | Euler polynomial | MATHEMATICS, APPLIED | BERNOULLI NUMBERS | INEQUALITIES | IDENTITIES | COMPLETE MONOTONICITY | STIRLING NUMBERS | POLYNOMIALS | EXPLICIT FORMULAS | INTEGRAL-REPRESENTATION | 2ND KIND | 1ST KIND

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 02/2011, Volume 217, Issue 12, pp. 5702 - 5728

Recently, the authors introduced some generalizations of the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials (see [Q.-M. Luo, H.M. Srivastava,...

Lerch’s functional equation | Hurwitz (or generalized), Hurwitz–Lerch and Lipschitz–Lerch zeta functions | Srivastava’s formula and Gaussian hypergeometric function | Genocchi numbers and Genocchi polynomials of higher order | Stirling numbers and the λ-Stirling numbers of the second kind | Apostol–Genocchi numbers and Apostol–Genocchi polynomials of higher order | Apostol–Bernoulli polynomials and Apostol–Euler polynomials of higher order | Apostol–Genocchi numbers and Apostol–Genocchi polynomials | Apostol-Bernoulli polynomials and Apostol-Euler polynomials of higher order | Srivastava's formula and Gaussian hypergeometric function | Hurwitz (or generalized), Hurwitz-Lerch and Lipschitz-Lerch zeta functions | Apostol-Genocchi numbers and Apostol-Genocchi polynomials | Apostol-Genocchi numbers and Apostol-Genocchi polynomials of higher order | Lerch's functional equation | FOURIER EXPANSIONS | MATHEMATICS, APPLIED | IDENTITIES | Q-EXTENSIONS | BERNOULLI | EXPLICIT FORMULA | ORDER | Stirling numbers and the lambda-Stirling numbers of the second kind | EULER POLYNOMIALS | INTEGRAL-REPRESENTATIONS | Hypergeometric functions | Analogue | Computation | Mathematical analysis | Gaussian | Mathematical models | Error correction | Representations

Lerch’s functional equation | Hurwitz (or generalized), Hurwitz–Lerch and Lipschitz–Lerch zeta functions | Srivastava’s formula and Gaussian hypergeometric function | Genocchi numbers and Genocchi polynomials of higher order | Stirling numbers and the λ-Stirling numbers of the second kind | Apostol–Genocchi numbers and Apostol–Genocchi polynomials of higher order | Apostol–Bernoulli polynomials and Apostol–Euler polynomials of higher order | Apostol–Genocchi numbers and Apostol–Genocchi polynomials | Apostol-Bernoulli polynomials and Apostol-Euler polynomials of higher order | Srivastava's formula and Gaussian hypergeometric function | Hurwitz (or generalized), Hurwitz-Lerch and Lipschitz-Lerch zeta functions | Apostol-Genocchi numbers and Apostol-Genocchi polynomials | Apostol-Genocchi numbers and Apostol-Genocchi polynomials of higher order | Lerch's functional equation | FOURIER EXPANSIONS | MATHEMATICS, APPLIED | IDENTITIES | Q-EXTENSIONS | BERNOULLI | EXPLICIT FORMULA | ORDER | Stirling numbers and the lambda-Stirling numbers of the second kind | EULER POLYNOMIALS | INTEGRAL-REPRESENTATIONS | Hypergeometric functions | Analogue | Computation | Mathematical analysis | Gaussian | Mathematical models | Error correction | Representations

Journal Article

Journal of Number Theory, ISSN 0022-314X, 12/2017, Volume 181, pp. 117 - 146

The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x;λ),...

Hypergeometric functions | Apostol–Bernoulli numbers and Apostol–Bernoulli polynomials | Humbert polynomials | Partial differential equations | Binomial coefficients | Generating functions | Apostol–Euler numbers and Apostol–Bernoulli polynomials | Daehee and Changhee numbers | Hurwitz–Lerch zeta functions | Cauchy numbers | Stirling numbers of the first kind | Functional equations | Lucas numbers | Hurwitz Lerch zeta functions | BERNOULLI NUMBERS | UNIFIED PRESENTATION | Apostol-Bernoulli polynomials | MATHEMATICS | Apostol-Euler numbers and | EULER POLYNOMIALS | Apostol-Bernoulli numbers and | 2ND KIND | FORMULAS | Medicine, Experimental | Medical research | Medical colleges | Statistics | Differential equations

Hypergeometric functions | Apostol–Bernoulli numbers and Apostol–Bernoulli polynomials | Humbert polynomials | Partial differential equations | Binomial coefficients | Generating functions | Apostol–Euler numbers and Apostol–Bernoulli polynomials | Daehee and Changhee numbers | Hurwitz–Lerch zeta functions | Cauchy numbers | Stirling numbers of the first kind | Functional equations | Lucas numbers | Hurwitz Lerch zeta functions | BERNOULLI NUMBERS | UNIFIED PRESENTATION | Apostol-Bernoulli polynomials | MATHEMATICS | Apostol-Euler numbers and | EULER POLYNOMIALS | Apostol-Bernoulli numbers and | 2ND KIND | FORMULAS | Medicine, Experimental | Medical research | Medical colleges | Statistics | Differential equations

Journal Article

Kyushu Journal of Mathematics, ISSN 1340-6116, 2013, Volume 67, Issue 1, pp. 143 - 153

We define the poly-Cauchy numbers by generalizing the Cauchy numbers of the first kind (the Bernoulli numbers of the second kind when divided by a factorial)....

poly-Cauchy numbers | Bernoulli numbers | Cauchy numbers | poly-Bernoulli numbers | Poly-bernoulli numbers | Poly-cauchy numbers | MATHEMATICS

poly-Cauchy numbers | Bernoulli numbers | Cauchy numbers | poly-Bernoulli numbers | Poly-bernoulli numbers | Poly-cauchy numbers | MATHEMATICS

Journal Article

HOKKAIDO MATHEMATICAL JOURNAL, ISSN 0385-4035, 10/2019, Volume 48, Issue 3, pp. 569 - 588

In this paper, we introduce the truncated Euler-Carlitz numbers as analogues of hypergeometric Euler numbers. In a special case, Euler-Carlitz numbers are...

MATHEMATICS | function fields | determinants | recurrence relations | Bernoulli-Carlitz numbers | Euler-Carlitz numbers | BERNOULLI-CARLITZ

MATHEMATICS | function fields | determinants | recurrence relations | Bernoulli-Carlitz numbers | Euler-Carlitz numbers | BERNOULLI-CARLITZ

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2019, Volume 2019, Issue 1, pp. 1 - 15

The Changhee numbers and polynomials are introduced by Kim, Kim and Seo (Adv. Stud. Theor. Phys. 7(20):993–1003, 2013), and the generalizations of those...

Fermionic p -adic q -integral on Z p ${\mathbb{Z}}_{p} | Analysis | Mathematics, general | ( h , q ) $(h,q)$ -Euler polynomials | Mathematics | Applications of Mathematics | Degenerate ( h , q ) $(h,q)$ -Changhee polynomials | Degenerate (h, q) -Changhee polynomials | (h, q) -Euler polynomials | Fermionic p-adic q-integral on Z | Q-EULER POLYNOMIALS | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | HIGHER-ORDER | IDENTITIES | (h, q)-Euler polynomials | Q-BERNOULLI | Degenerate (h, q)-Changhee polynomials | Fermionic p-adic q-integral on Z(p) | Polynomials | Fermionic p-adic q-integral on Z p ${\mathbb{Z}}_{p}

Fermionic p -adic q -integral on Z p ${\mathbb{Z}}_{p} | Analysis | Mathematics, general | ( h , q ) $(h,q)$ -Euler polynomials | Mathematics | Applications of Mathematics | Degenerate ( h , q ) $(h,q)$ -Changhee polynomials | Degenerate (h, q) -Changhee polynomials | (h, q) -Euler polynomials | Fermionic p-adic q-integral on Z | Q-EULER POLYNOMIALS | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | HIGHER-ORDER | IDENTITIES | (h, q)-Euler polynomials | Q-BERNOULLI | Degenerate (h, q)-Changhee polynomials | Fermionic p-adic q-integral on Z(p) | Polynomials | Fermionic p-adic q-integral on Z p ${\mathbb{Z}}_{p}

Journal Article

Mediterranean Journal of Mathematics, ISSN 1660-5446, 06/2017, Volume 14, Issue 3, p. 1

In this paper we introduce a q-analogue of the incomplete poly-Bernoulli numbers and incomplete poly-Cauchy numbers by using the q-Hurwitz-Lerch zeta Function....

generating function | incomplete poly-Cauchy numbers | q-Hurwitz–Lerch zeta function | combinatorial identities | Incomplete poly-Bernoulli numbers | MATHEMATICS | MATHEMATICS, APPLIED | q-Hurwitz-Lerch zeta function

generating function | incomplete poly-Cauchy numbers | q-Hurwitz–Lerch zeta function | combinatorial identities | Incomplete poly-Bernoulli numbers | MATHEMATICS | MATHEMATICS, APPLIED | q-Hurwitz-Lerch zeta function

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 10/2018, Volume 334, pp. 288 - 294

The main purpose of this paper is using the elementary method and the properties of trigonometric functions to study the computational problem of one kind...

Dirichlet L-functions | Trigonometric sums | Computational formula | Bernoulli numbers | Identity

Dirichlet L-functions | Trigonometric sums | Computational formula | Bernoulli numbers | Identity

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2015, Volume 2015, Issue 1, pp. 1 - 9

The p-adic q-integral (sometimes called q-Volkenborn integration) was defined by Kim. From p-adic q-integral equations, we can derive various q-extensions of...

(h, q)-Daehee numbers | p-adic q-integral | (h, q)-Bernoulli polynomials | (h, q)-Daehee polynomials | MATHEMATICS | MATHEMATICS, APPLIED | Q-BERNOULLI NUMBERS | DAEHEE | Integral equations | Analysis | Polynomials | Texts | Difference equations | Mathematical analysis

(h, q)-Daehee numbers | p-adic q-integral | (h, q)-Bernoulli polynomials | (h, q)-Daehee polynomials | MATHEMATICS | MATHEMATICS, APPLIED | Q-BERNOULLI NUMBERS | DAEHEE | Integral equations | Analysis | Polynomials | Texts | Difference equations | Mathematical analysis

Journal Article

Journal of Integer Sequences, 2018, Volume 21, Issue 6

Journal Article

Monatshefte für Mathematik, ISSN 0026-9255, 6/2016, Volume 180, Issue 2, pp. 271 - 288

By using the restricted and associated Stirling numbers of the first kind by generalizing the (unsigned) Stirling numbers of the first kind, we define the...

Restricted Stirling numbers | Secondary 11B75 | Restricted poly-Cauchy numbers | Mathematics, general | Associated Stirling numbers | 05A19 | Mathematics | Primary 11B73 | Associated poly-Cauchy numbers | 05A15 | Poly-Cauchy numbers | MATHEMATICS | BERNOULLI NUMBERS

Restricted Stirling numbers | Secondary 11B75 | Restricted poly-Cauchy numbers | Mathematics, general | Associated Stirling numbers | 05A19 | Mathematics | Primary 11B73 | Associated poly-Cauchy numbers | 05A15 | Poly-Cauchy numbers | MATHEMATICS | BERNOULLI NUMBERS

Journal Article

Journal of the Australian Mathematical Society, ISSN 1446-7887, 11/2016, Volume 103, Issue 1, pp. 1 - 19

Poly-Euler numbers are introduced as a generalization of the Euler numbers in a manner similar to the introduction of the poly-Bernoulli numbers. In this...

poly-Bernoulli number | Clausen–von Staudt theorem | Euler number | poly-Euler number | MATHEMATICS | BERNOULLI NUMBERS | Clausen-von Staudt theorem

poly-Bernoulli number | Clausen–von Staudt theorem | Euler number | poly-Euler number | MATHEMATICS | BERNOULLI NUMBERS | Clausen-von Staudt theorem

Journal Article

Ars Combinatoria, ISSN 0381-7032, 01/2018, Volume 136, pp. 199 - 210

We give convolution identities for Tribonacci numbers without binomial coefficients and with binomial coefficients.

MATHEMATICS | BERNOULLI

MATHEMATICS | BERNOULLI

Journal Article

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