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74
The 2adic valuation of the coefficients of a polynomial
 Scientia, Series A
"... Abstract. In this paper we compute the 2adic valuations of some polynomials associated with the definite integral ∫ ∞ dx 0 (x4 + 2ax2. + 1) m+1 1. ..."
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Abstract. In this paper we compute the 2adic valuations of some polynomials associated with the definite integral ∫ ∞ dx 0 (x4 + 2ax2. + 1) m+1 1.
A Finite Difference Approach To Degenerate Bernoulli And Stirling Polynomials
 DISCRETE MATH
, 1992
"... Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Some particular calculati ..."
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Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Some particular calculations are done and several factorization results are proven and conjectured.
The asymptotic behavior of the Stirling numbers of the first kind
"... this paper we will first prove the theorem above, then extend it to a form from which one can find as many terms of the expansion as desired, and finally discuss the relationship of our work to earlier results of Moser and Wyman [3]. * Research supported by the United States Office of Naval Research ..."
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this paper we will first prove the theorem above, then extend it to a form from which one can find as many terms of the expansion as desired, and finally discuss the relationship of our work to earlier results of Moser and Wyman [3]. * Research supported by the United States Office of Naval Research. 2. Proof of theorem 1 We will write i n (s) for P n j=1 j \Gammas , which, if s ? 1, is the nth partial sum of the Dirichlet series for the Riemann zeta function. The following computation will be the basis for the developments in the sequel. We have, from the familiar generating function, = (1 + x)(1 + ) \Delta \Delta \Delta (1 + log(1 + )g sj exp f(\Gamma1) g: (1) In the last member we wish to replace all of the i n 's by their asymptotic expansions in powers of n and log n. These expansions are (e.g., [5], p. 124) i n (s) log n + fl + d(1; j)n \Gammaj ; if s = 1; i(s) + (1 \Gamma s) \Gammas+1 j=1 d(s; j)n \Gammas\Gammaj+1 ; if s ? 1, (2) where d(s; j) = \Gamma B j s + j \Gamma 2 and the B j 's are the Bernoulli numbers. By combining (1) and (2), we now have exp x(log n + fl + d(1; j) \Theta i(s) + 1 (1 \Gamma s)n d(s; j) s+j \Gamma1 (3) in the sense that for each fixed k, the coefficient of x on the left has, for its complete asymptotic series as n !1, the coefficient of x on the right. Now we can prove theorem 1. To do that we need only keep the portion of the development in powers of n and log n that does not approach zero on the right side of (3), along with the size of the first neglected term. If we take the coefficient of x on both sides of (3) we obtain x(log n+fl) \Psi\Psi The second exponential factor above is ...
Congruences and recurrences for Bernoulli numbers of higher order, Fibonacci Quart. 32
 MR 95k: 11021
, 1994
"... The Bernoulli polynomials of order k, for any integer k, may be defined by (see [10], p. 145): x V 2 ^ ^ w, * " = y # ( z). (i.i) ..."
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The Bernoulli polynomials of order k, for any integer k, may be defined by (see [10], p. 145): x V 2 ^ ^ w, * " = y # ( z). (i.i)
COMBINATORIAL MODELS OF CREATION–ANNIHILATION
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65C
, 2011
"... Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator ..."
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Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or “diagrams”, that are composed of elementary “gates”. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to qanalogues, multivariate frameworks, and urn models are also briefly discussed.
The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions
 Mem. Amer. Math. Soc
"... Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, ..."
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Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of rightpointing and leftpointing unit triangles in each hole. We detail this parallel by indicating that, as a consequence of our result, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approaches, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. We give an equivalent phrasing of our result in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatics arises by averaging over all possible discrete geometries of the covering surfaces.
Algorithms for Bernoulli and allied polynomials
 J. Integer Seq
"... We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers ..."
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We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind. 1
On Differences of Zeta Values
 in "Journal of Computational and Applied Mathematics
, 2008
"... ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of Nörlu ..."
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ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of NörlundRice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li’s criterion for the Riemann hypothesis.