Results 21 to 30 of about 10,813 (266)
Congruences for Bernoulli numbers and Bernoulli polynomials
Discrete Mathematics, 1997 The Bernoulli numbers and polynomials are defined by \(B_0=1\), \(\sum^{n-1}_{k=0}{n\choose k} B_k= 0\) \((n=2,3,\dots)\) and \(B_n(x)= \sum^n_{k=0}{n\choose k} B_{n-k} x^k\), respectively. Two basic congruences for Bernoulli numbers are the Kummer congruences (used in the theory of Fermat's last theorem) and the von Staudt-Clausen theorem. There exist
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q-Bernoulli Numbers Associated with q-Stirling Numbers
Advances in Difference Equations, 2008 We consider Carlitz q-Bernoulli numbers and q-Stirling numbers of the first and the second kinds. From the properties of q-Stirling numbers, we derive many interesting formulas associated with Carlitz q-Bernoulli numbers. Finally, we will prove βn,q=â
Taekyun Kim
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Demonstratio Mathematica, 2022 In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of ...
Chen Xue-Yan +3 more
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Advances in Difference Equations, 2010 We define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials attached to χ of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain
L.-C. Jang +5 more
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Advanced Materials, EarlyView.We present elastomeric three‐dimensional (3D) microstructures fabricated via two‐photon polymerization (2PP) and post‐processed through wet etching, for quantifying nanonewton (nN)‐scale forces applied by healthy and diseased neural cells. The mechanically characterized free‐standing beam architectures enable measurement of traction forces of ...
Pieter F. J. van Altena +7 more
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Explicit Formulas Involving -Euler Numbers and Polynomials
Abstract and Applied Analysis, 2012 We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator.
Serkan Araci +2 more
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Mathematics, 2021 In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions,
Daeyeoul Kim, Yilmaz Simsek
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Journal of Number Theory, 1995 The author strengthens the Sylvester-Lipschitz theorem for Bernoulli numbers \(B_m\) as follows: ``For an integer \(a\) and a positive integer \(m\) the number \(a^{[\log_2 m]+1} (a^m- 1)B_m/ m\) is an integer.'' It is noted that in a certain sense this strengthening of the Sylvester- Lipschitz theorem is the best possible.
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Bernoulli Related Polynomials and Numbers [PDF]
Mathematics of Computation, 1979 The polynomials φ
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A Review on Sensor Technologies, Control Approaches, and Emerging Challenges in Soft Robotics
Advanced Robotics Research, EarlyView.This review provides an introspective of sensors and controllers in soft robotics. Initially describing the current sensing methods, then moving on to the control methods utilized, and finally ending with challenges and future directions in soft robotics focusing on the material innovations, sensor fusion, and embedded intelligence for sensors and ...
Ean Lovett +5 more
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