Results 11  20
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102
Lattice Paths between Diagonal Boundaries
 Electronic J. Combinatorics
, 1998
"... A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n1])+# 1 (d[mr 1 ,ns 1 ]+d[ms 1 ,nr 1 ])++# k (d[mr k ,ns k ] +d[ms k ,nr k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetri ..."
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A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n1])+# 1 (d[mr 1 ,ns 1 ]+d[ms 1 ,nr 1 ])++# k (d[mr k ,ns k ] +d[ms k ,nr k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x + u<y<xl. With a solution we mean a formula that expresses d[m, n] as a sum of di#erences of recursions without the band restriction. Depending on the application, the boundary conditions can take di#erent forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x # 0, and d[x l, x] = 0 for all x # l (applies to the exact distribution of the KolmogorovSmirnov twosample statistic), d[x + u, x]=0 for all x # 0, and d[x  l +1,x]=d[xl+1,x1] for x # l (ordinary lattice paths with weighted left turns), and d[y, y  u +1]=d[y1,yu+1]for all y # u and d[x  l +1,x]=d[xl+1,x1] for x # l. The first theorem is a gene...
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 ...
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.
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)
From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
, 707
"... This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. T ..."
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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is
Torney, A symbolic operator approach to several summation formulas for power series
 J. Comp. Appl. Math
"... This paper deals with the summation problem of power series of the form Sba(f;x) = a≤k≤b f(k)x k, where 0 ≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expans ..."
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This paper deals with the summation problem of power series of the form Sba(f;x) = a≤k≤b f(k)x k, where 0 ≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for Sba(f;x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.
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.