Bernoulli numbers and polynomialsa bibliography of
Bernoulli numbers
"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" S.C. Woon, "A new representation of the Riemann zeta function ζ(s)" [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."
A. Kuznetsov, "Nontrivial zeros of the Riemann zeta function as the limit of eigenvalues of nonsymmetric matrices" (preprint 2006) [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."
J.P. Edwards, C. Moctezuma Mata and C. Schubert, "One-loop amplitudes in the worldline formalism", Physica Scripta 97 (2022) 064002 [summary:] The worldline formalism packages whole families of one-loop Feynman diagrams into master integrals. For low-energy N-photon amplitudes in scalar and spinor QED, and for general one-loop N-point integrals in scalar φ3 theory, the calculation is organised by inverse derivatives on periodic functions whose kernels are Bernoulli polynomials – making Bernoulli numbers and polynomials central rather than incidental to the algorithm.
A. Padilla and R.G.C. Smith, "Smoothed asymptotics: From number theory to QFT", Physical Review D 110 (2024) 025010 [summary:] Beginning with smoothed asymptotics for divergent sums of powers, where the finite terms are encoded by Bernoulli numbers and the values ζ(−n), an ultraviolet scheme called eta regularisation is constructed for loop integrals. Applied to non-Abelian gauge fields coupled to Dirac fermions, suitable eta regulators preserve the Ward identity for the one-loop vacuum-polarisation tensor.
R. Bittleston and K. Costello, "The One-Loop QCD Beta Function as an Index", Physical Review Letters 136 (2026) 211601 [summary:] The one-loop QCD beta function is recovered from an index theorem after self-dual gauge theory is rewritten as a holomorphic theory on twistor space. In the index calculation the characteristic-class expansion produces Bernoulli-number factors, giving a direct route from the arithmetic coefficients in the Todd series to familiar perturbative gauge-theory constants.
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."
M.W. Coffey, "Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant" (preprint 06/2011) "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."
G. Ottarsson, "The Ladder Hypothesis" [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.
zeta
functions related to poly-Bernoulli numbers
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