New asymptotic expansions on hyperfactorial functions by Xu, Jianjun

“…In this paper, by using the Bernoulli numbers and the exponential complete Bell polynomials, we establish four general asymptotic expansions for the hyperfactorial functions $\prod _{k=1}^n {k^{k^q}}$, which have only odd power terms or even power terms. …”
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On the Order of Certain Characteristic Classes of the Hodge Bundle of Semi-Abelian Schemes by Maillot, V, Roessler, D

“…These bounds appear in the numerators of modified Bernoulli numbers. We also obtain similar results in an equivariant situation.…”

A Parametric Type of Cauchy Polynomials with Higher Level by Takao Komatsu

“…Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. …”
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Degenerate Cauchy numbers of the third kind by Sung-Soo Pyo, Taekyun Kim, Seog-Hoon Rim

“…In addition, we give some relations between four kinds of the degenerate Cauchy numbers, the Daehee numbers and the degenerate Bernoulli numbers.…”
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Periodic solutions for a higher-order p-Laplacian neutral differential equation with multiple deviating arguments by Loubna Moutaouekkil, Omar Chakrone, Zakaria El Allali, Said Taarabti

“….$$ By appling the continuation theorem, theory of Fourier series, Bernoulli numbers theory and some analytic techniques, sufficient conditions for the existence of periodic solutions are established. …”
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On application and analysis of helicoidal shells in architecture and civil engineering by M I Rynkovskaya

“…Krivoshapko was simplified by author due to the application of Bernoulli numbers in the process of integration of equations. …”
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Extended p-adic q-invariant integrals on ℤp associated with applications of umbral calculus by Araci, Serkan, Acikgoz, Mehmet, Kilicman, Adem

“…From those considerations, we derive some new interesting properties on the extended p-adic q-Bernoulli numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the p-adic q-Bernoulli polynomials are given in the present work.…”
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Distributional Analysis of the Parking Problem and Robin Hood Linear Probing Hashing with Buckets by Alfredo Viola

“…A key element in the analysis is the use of a new family of numbers, called Tuba Numbers, that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.…”
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Note on the Higher-Order Derivatives of the Hyperharmonic Polynomials and the <em>r</em>-Stirling Polynomials of the First Kind by José L. Cereceda

“…In addition, we obtain various identities involving the <i>r</i>-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the Stirling numbers of the first and second kind, and the harmonic numbers.…”
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Distributional analysis of Robin Hood linear probing hashing with buckets by Alfredo Viola

“…A key element in the analysis is the use of a new family of numbers that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.…”
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Dirichlet series under standard convolutions: variations on Ramanujan’s identity for odd zeta values by Chavan, Parth, Chavan, Sarth, Vignat, Christophe, Wakhare, Tanay

“…Abstract Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by Dixit et al., or the Fourier expansion of the convolution of Bernoulli–Barnes polynomials by Komori et al.…”
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A Note on Degenerate Catalan-Daehee Numbers and Polynomials by Waseem Ahmad Khan, Maryam Salem Alatawi, Ugur Duran

“…Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan–Daehee numbers and Stirling numbers of the second kind. …”
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Can a Wind Model Mimic a Convection-Dominated Accretion Flow Model? by Heon-Young Chang

“…In this paper we investigate the properties of advection-dominated accretion flows(ADAFs) in case that outflows carry away infalling matter with its angular momentum and energy. Positive Bernoulli numbers in ADAFs allow a fraction of the gas to be ex-pelled in a form of outflows. …”
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Some Identities with Multi-Generalized <i>q</i>-Hyperharmonic Numbers of Order <i>r</i> by Zhihua Chen, Neşe Ömür, Sibel Koparal, Waseem Ahmad Khan

“…Additionally, one of the applications is the sum involving <i>q</i>-Stirling numbers and <i>q</i>-Bernoulli numbers.…”
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Starlike Functions Associated with Bernoulli’s Numbers of Second Kind by Mohsan Raza, Mehak Tariq, Jong-Suk Ro, Fairouz Tchier, Sarfraz Nawaz Malik

“…</mo></mrow></mfrac></mrow></semantics></math></inline-formula>, where the coefficients of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>B</mi><mrow><mi>n</mi></mrow><mn>2</mn></msubsup></semantics></math></inline-formula> are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>𝟊</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><mi>ξ</mi><mi>𝟊</mi><mo>′</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mrow><mrow><mi>𝟊</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mfrac><mo>≺</mo><msub><mi>φ</mi><mrow><mi>B</mi><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>.…”
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Diverse Properties and Approximate Roots for a Novel Kinds of the (<i>p</i>,<i>q</i>)-Cosine and (<i>p</i>,<i>q</i>)-Sine Geometric Polynomials by Sunil Kumar Sharma, Waseem Ahmad Khan, Cheon-Seoung Ryoo, Ugur Duran

“…Utilizing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced></semantics></math></inline-formula>-numbers and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced></semantics></math></inline-formula>-concepts, in 2016, Duran et al. considered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced></semantics></math></inline-formula>-Genocchi numbers and polynomials, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced></semantics></math></inline-formula>-Bernoulli numbers and polynomials and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced></semantics></math></inline-formula>-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. …”
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New Bounds for Arithmetic Mean by the Seiffert-like Means by Ling Zhu

“…By using the power series of the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mo form="prefix">sin</mo><mi>n</mi></msup><mi>t</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mo form="prefix">cos</mo><mi>t</mi></mfenced><mo>/</mo><mfenced separators="" open="(" close=")"><msup><mo form="prefix">sin</mo><mi>n</mi></msup><mi>t</mi></mfenced></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></semantics></math></inline-formula>), and the estimation of the ratio of two adjacent Bernoulli numbers, we obtained new bounds for arithmetic mean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> by the weighted arithmetic means of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">M</mi><mrow><mo form="prefix">tan</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><msubsup><mi mathvariant="script">M</mi><mrow><mo form="prefix">sin</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi mathvariant="script">M</mi><mo form="prefix">tan</mo></msub><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi mathvariant="script">M</mi><mo form="prefix">sin</mo></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">M</mi><mrow><mo form="prefix">tanh</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><msubsup><mi mathvariant="script">M</mi><mrow><mo form="prefix">sinh</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msub><mi mathvariant="script">M</mi><mo form="prefix">tanh</mo></msub><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi mathvariant="script">M</mi><mo form="prefix">sinh</mo></msub><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mo form="prefix">tan</mo></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mo form="prefix">sin</mo></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mo form="prefix">tanh</mo></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mo form="prefix">sinh</mo></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are the tangent mean, sine mean, hyperbolic tangent mean and hyperbolic sine mean, respectively. …”
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