2009, ISBN 9780195334548, xiv, 422

Like the intriguing Fibonacci and Lucas numbers, Catalan numbers are also ubiquitous. "They have the same delightful propensity for popping up unexpectedly,...

Catalan numbers (Mathematics) | Mathematics | Euler's triangulation problem | Pascal's triangle | Pascal's identity | Lucas numbers | Catalan sequence | Parenthesization problem | Catalan numbers | Martin gardner | Fibonacci numbers

Catalan numbers (Mathematics) | Mathematics | Euler's triangulation problem | Pascal's triangle | Pascal's identity | Lucas numbers | Catalan sequence | Parenthesization problem | Catalan numbers | Martin gardner | Fibonacci numbers

Book

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

American Journal of Physiology - Lung Cellular and Molecular Physiology, ISSN 1040-0605, 2015, Volume 309, Issue 11, pp. L1286 - L1293

The lung parenchyma provides a maximal surface area of blood-containing capillaries that are in close contact with a large surface area of the air-containing...

Euler number | Stereology | Capillary number | capillary number | PHYSIOLOGY | PULMONARY-HYPERTENSION | SURFACE-AREA | HUMAN LUNG | LENGTH | VOLUME | stereology | RAT LUNG | GLOMERULAR CAPILLARIES | RESPIRATORY SYSTEM | ARCHITECTURE | UNBIASED ESTIMATION | MAMMALIAN LUNG | Animals | Capillaries - ultrastructure | Cell Count | Rats | Physiology - methods | Capillaries - anatomy & histology | Imaging, Three-Dimensional | Pulmonary Alveoli - blood supply | Measurement | Physiological aspects | Lungs | Capillaries

Euler number | Stereology | Capillary number | capillary number | PHYSIOLOGY | PULMONARY-HYPERTENSION | SURFACE-AREA | HUMAN LUNG | LENGTH | VOLUME | stereology | RAT LUNG | GLOMERULAR CAPILLARIES | RESPIRATORY SYSTEM | ARCHITECTURE | UNBIASED ESTIMATION | MAMMALIAN LUNG | Animals | Capillaries - ultrastructure | Cell Count | Rats | Physiology - methods | Capillaries - anatomy & histology | Imaging, Three-Dimensional | Pulmonary Alveoli - blood supply | Measurement | Physiological aspects | Lungs | Capillaries

Journal Article

2018, First edition., ISBN 0198794924, 162 pages

Book

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 | (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 | (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

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

Annales de l'Institut Fourier, ISSN 0373-0956, 2012, Volume 62, Issue 6, pp. 2315 - 2345

Journal Article

2006, New ed., Princeton Science Library, ISBN 0691118221, xx, 380

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research,...

Numbers, Complex | Mathematics | History of Science & Technology | History | Euler’s numbers | Euler's numbers

Numbers, Complex | Mathematics | History of Science & Technology | History | Euler’s numbers | Euler's numbers

Book

Journal of Number Theory, ISSN 0022-314X, 2011, Volume 131, Issue 12, pp. 2387 - 2397

The nth Delannoy number and the nth Schröder number given by D n = ∑ k = 0 n ( n k ) ( n + k k ) and S n = ∑ k = 0 n ( n k ) ( n + k k ) 1 k + 1 respectively...

Schröder numbers | Central Delannoy numbers | Congruences | Euler numbers | MATHEMATICS | Schroder numbers

Schröder numbers | Central Delannoy numbers | Congruences | Euler numbers | MATHEMATICS | Schroder numbers

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2018, Volume 466, Issue 1, pp. 1009 - 1042

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic...

Euler sums | Riemann zeta function | Multiple zeta (star) values | Stirling numbers | Harmonic numbers | Multiple harmonic (star) numbers | INTEGRALS | MATHEMATICS | MULTIPLE ZETA-VALUES | MATHEMATICS, APPLIED | SERIES

Euler sums | Riemann zeta function | Multiple zeta (star) values | Stirling numbers | Harmonic numbers | Multiple harmonic (star) numbers | INTEGRALS | MATHEMATICS | MULTIPLE ZETA-VALUES | MATHEMATICS, APPLIED | SERIES

Journal Article

Applicable Analysis and Discrete Mathematics, ISSN 1452-8630, 10/2019, Volume 13, Issue 2, pp. 478 - 494

In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for...

MATHEMATICS | Apostol-type numbers and polynomials | MATHEMATICS, APPLIED | Generating functions | Combinatorial numbers | BERNOULLI | Stirling numbers | EULER | Computation algorithm

MATHEMATICS | Apostol-type numbers and polynomials | MATHEMATICS, APPLIED | Generating functions | Combinatorial numbers | BERNOULLI | Stirling numbers | EULER | Computation algorithm

Journal Article

2017, First edition., ISBN 9780465093779, viii, 221 pages

"Bertrand Russell wrote that mathematics can exalt "as surely as poetry." This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard...

Mathematics | Euler's numbers | Euler, Leonhard, 1707-1783 | History | Numbers, Complex

Mathematics | Euler's numbers | Euler, Leonhard, 1707-1783 | History | Numbers, Complex

Book

Journal of Beijing Institute of Technology (English Edition), ISSN 1004-0579, 09/2015, Volume 24, Issue 3, pp. 298 - 304

Journal Article

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, ISSN 1578-7303, 1/2019, Volume 113, Issue 1, pp. 281 - 297

The aim of this paper is to construct interpolation functions for the numbers of the k-ary Lyndon words which count n digit primitive necklace class...

Lyndon words | 03D40 | Arithmetical functions | Special polynomials | Frobenius–Euler numbers and polynomials | 11S40 | Theoretical, Mathematical and Computational Physics | 11M35 | Generating functions | 68R15 | Mathematics | Algorithm | 11A25 | 11B68 | Stirling numbers of the first kind | 47E05 | 11B83 | Mathematics, general | Apostol–Euler numbers and polynomials | Differential operator | Applications of Mathematics | Special numbers | 05A05 | 05A15 | MATHEMATICS | Apostol-Euler numbers and polynomials | BERNOULLI | Frobenius-Euler numbers and polynomials | Operators (mathematics) | Interpolation | Algorithms | Infinite series | Combinatorial analysis | Sums

Lyndon words | 03D40 | Arithmetical functions | Special polynomials | Frobenius–Euler numbers and polynomials | 11S40 | Theoretical, Mathematical and Computational Physics | 11M35 | Generating functions | 68R15 | Mathematics | Algorithm | 11A25 | 11B68 | Stirling numbers of the first kind | 47E05 | 11B83 | Mathematics, general | Apostol–Euler numbers and polynomials | Differential operator | Applications of Mathematics | Special numbers | 05A05 | 05A15 | MATHEMATICS | Apostol-Euler numbers and polynomials | BERNOULLI | Frobenius-Euler numbers and polynomials | Operators (mathematics) | Interpolation | Algorithms | Infinite series | Combinatorial analysis | Sums

Journal Article

Journal of Number Theory, ISSN 0022-314X, 12/2012, Volume 132, Issue 12, pp. 2854 - 2865

In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our...

Sums product of Euler numbers | Euler numbers | Frobenius–Euler numbers and polynomials | Differential equations | Frobenius-Euler numbers and polynomials | MATHEMATICS | HIGHER-ORDER | NUMBERS

Sums product of Euler numbers | Euler numbers | Frobenius–Euler numbers and polynomials | Differential equations | Frobenius-Euler numbers and polynomials | MATHEMATICS | HIGHER-ORDER | NUMBERS

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2011, Volume 54, Issue 12, pp. 2509 - 2535

Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is...

欧拉数 | 组合性质 | Euler数 | 超几何级数 | 同余 | Jacobi符号 | 一致性 | 伯努利数 | 11A07 | 11E25 | super congruences | Euler numbers | 05A10 | 33C20 | 11B65 | Mathematics | 11B68 | 11F20 | central binomial coefficients | 05A19 | 11S99 | 11M06 | Applications of Mathematics | 1/PI | MATHEMATICS | MATHEMATICS, APPLIED | GAUSSIAN HYPERGEOMETRIC-SERIES | BERNOULLI NUMBERS | SUPERCONGRUENCE CONJECTURE | Congruences | Mathematical analysis | China | Combinatorial analysis | Sums

欧拉数 | 组合性质 | Euler数 | 超几何级数 | 同余 | Jacobi符号 | 一致性 | 伯努利数 | 11A07 | 11E25 | super congruences | Euler numbers | 05A10 | 33C20 | 11B65 | Mathematics | 11B68 | 11F20 | central binomial coefficients | 05A19 | 11S99 | 11M06 | Applications of Mathematics | 1/PI | MATHEMATICS | MATHEMATICS, APPLIED | GAUSSIAN HYPERGEOMETRIC-SERIES | BERNOULLI NUMBERS | SUPERCONGRUENCE CONJECTURE | Congruences | Mathematical analysis | China | Combinatorial analysis | Sums

Journal Article

Journal of the Korean Mathematical Society, ISSN 0304-9914, 2017, Volume 54, Issue 5, pp. 1605 - 1621

The purpose of this paper is to construct a new family of the special numbers which are related to the Fubini type numbers and the other well-known special...

Gen-erating functions | Combinatorial sum | Frobenius-Euler numbers | Bernoulli numbers | Binomial coefficients | Functional equations | Fubini numbers | Stirling numbers | Apostol-Bernoulli numbers | Apostol-Bernoulli polynomials | MATHEMATICS, APPLIED | IDENTITIES | binomial coefficients | generating functions | UNIFIED PRESENTATION | combinatorial sum | MATHEMATICS | ZETA | functional equations | ApostolBernoulli polynomials | UNIFICATION | EULER

Gen-erating functions | Combinatorial sum | Frobenius-Euler numbers | Bernoulli numbers | Binomial coefficients | Functional equations | Fubini numbers | Stirling numbers | Apostol-Bernoulli numbers | Apostol-Bernoulli polynomials | MATHEMATICS, APPLIED | IDENTITIES | binomial coefficients | generating functions | UNIFIED PRESENTATION | combinatorial sum | MATHEMATICS | ZETA | functional equations | ApostolBernoulli polynomials | UNIFICATION | EULER

Journal Article

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

Proceedings of the Royal Society B: Biological Sciences, ISSN 0962-8452, 02/2007, Volume 274, Issue 1609, pp. 599 - 604

Mathematical models of transmission have become invaluable management tools in planning for the control of emerging infectious diseases. A key variable in such...

Reproductive tract infections | Mathematical intervals | Epidemics | Musical intervals | Infectious diseases | Disease models | Generating function | Influenza A virus | Infections | Mathematical moments | Serial interval | Epidemiology | Influenza | Lotka-Euler equation | Basic reproduction ratio | basic reproduction ratio | influenza | epidemiology | EVOLUTIONARY BIOLOGY | BIOLOGY | DYNAMICS | serial interval | ECOLOGY | EPIDEMIC | STRATEGIES | Models, Biological | Humans | Disease Outbreaks - prevention & control | Influenza, Human - epidemiology | Cohort Effect | Influenza, Human - transmission | Influenza, Human - prevention & control | Lotka–Euler equation

Reproductive tract infections | Mathematical intervals | Epidemics | Musical intervals | Infectious diseases | Disease models | Generating function | Influenza A virus | Infections | Mathematical moments | Serial interval | Epidemiology | Influenza | Lotka-Euler equation | Basic reproduction ratio | basic reproduction ratio | influenza | epidemiology | EVOLUTIONARY BIOLOGY | BIOLOGY | DYNAMICS | serial interval | ECOLOGY | EPIDEMIC | STRATEGIES | Models, Biological | Humans | Disease Outbreaks - prevention & control | Influenza, Human - epidemiology | Cohort Effect | Influenza, Human - transmission | Influenza, Human - prevention & control | Lotka–Euler equation

Journal Article

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