## Bernoulli numbers and polynomials

"The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenomenon of 'scatterings' of the zeros of [the Bernoulli polynomials] Bs(z) is observed"

[abstract:] "A generalization of a well-known relation between the Riemann zeta function representation of the Riemann zeta function in terms of a nested series of Bernoulli numbers."

C. Musès, "Applied hypernumbers: computational concepts", Applied Mathematics and Computation 3 (1977) 211-226

[abstract:] "The key importance of hypernumbers in enlarging and fruitfully generalizing (as distinct from abstraction of a sterile sort) algebra, function theory and computation is discussed, with specific examples and theorems. The rich serendipity of hypernumber research is shown in the author's recent findings; for example, those generalizing the Bernoulli numbers for any real, complex, or countercomplex index s, as $B_s = -s!2\cos(\pi s/2) \zeta(s)/(2\pi)^{s}$, where $\zeta$ is Riemann's Zeta function; whence, e.g., B0 =1, B2 = 1/6, B1/2 = 1/2 zeta(1/2), and B3/B5 = - (802)-1zeta(3)/zeta(5), results like the last two being unknown and unobtainable before. As in APL computer language, the symbol "!" is used to denote Gauss' function: the factorial of unrestricted argument."

C. Musès, "Some new considerations on the Bernoulli numbers, the factorial function, and Riemann's zeta function", Applied Mathematics and Computation 113 (2000) 1-21.

Curiously, although published, this includes a supposed proof of the RH on page 21. Musès died around the time of publication. You should be able to view a PDF version of this article at this site (just log in as a guest).

P.R. Subramanian, "Generating functions for angular momentum traces", Journal of Physics A 19 (1986) 2667-2670.

[Abstract:] "Generating functions for Tr(Jlambda2p) are obtained, one of them being the character of a representation of the three-dimensional pure rotation group. Recurrence relations for the Bernoulli numbers and the Riemann zeta functions are deduced."

P.R. Subramanian, "Evaluation of Tr(Jlambda2p) using the Brillouin function", Journal of Physics A 19 (1986) 1179-1187.

[Abstract:] "Obtains expressions for Tr(Jlambda2p) in terms of the Brillouin function. Standard properties of Tr(Jlambda2p) are derived from them. Sum rules for the Bernoulli numbers and the Riemann zeta functions are deduced as corollaries."

[abstract:] "We construct a family of $2n \times 2n$ matrices $B_{2n}$, such that the spectrum of $\sigma(B_{2n})$ converges to the set of nontrivial zeros of the Riemann zeta function $\zeta(s)$. The coefficients of these matrices are given explicitly as finite sums of Bernoulli numbers."

M. de Gosson, B. Dragovich and A. Khrennikov, "Some p-adic differential equations"

"We investigate various properties of p-adic differential equations which have as a solution an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k (n) x^n$, where $P_k (n) = n^k + C_{k-1} n^{k-1} + ...+ C_0$ is a polynomial in n with $C_i\in Z$ (in a more general case $C_i\in Q$ or $C_i\in C_p$). For some special classes of $P_k (n)$, as well as for the general case, the existence of the corresponding linear differential equations of the first- and second-order for $F_k (x)$, is shown. In some cases such equations are constructed. For the second-order differential equations there is no other analytic solution of the form $\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous first-order differential equation exists one can construct infinitely many inhomogeneous second-order equations with the same analytic solution. Relation to some rational sums with the Bernoulli numbers and to $F_k (x)$ for some $x\in Z$ is considered. Some of these differential equations can be related to p-adic dynamics and p-adic information theory."

G. Everest, Y. Puri and T. Ward, "Integer sequences counting periodic points"

[Abstract:] "An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the p-parts of an integer sequence themselves count periodic points are studied. The Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide another. The methods give a dynamical interpretation of many classical congruences such as Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied by the Bernoulli numbers."

A. Veselov and J. Ward, "On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function" (1999)

U. Muller and C. Schubert, "A quantum field theoretical representation of Euler-Zagier sums"

[abstract:] "We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of an algebra of elementary vertex integrals whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of the model are pointed out."

"We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $\zeta(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $\gamma_1(a)$. The presentation unifies some earlier results."

[Abstract:] "In a paper from 18 August 2001 available at www.islandia.is/gko/010818.pdf, a thermoelectric generator was constructed from a large number of series connected parallelepipeds. The hot and/or cold reservoir was made of some electrically conductive metal, and the fluid was to some extent conductive to the electrical ground. This topology generated a number of small capacitors, each formed by two parallelepiped crystal faces and the grounded thermal reservoir. When analysing the frequency behaviour of such a device, rational polynomials manifested themselves and proved to be a rich source of advanced mathematical relations."

These relations involve the Riemann zeta function, Bernoulli numbers, the Gamma function, Euler's constant and Stirling's Formula.

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