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The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function
"... Abstract. The table of Gradshteyn and Rhyzik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some formulas in the mentioned table. 1. ..."
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Abstract. The table of Gradshteyn and Rhyzik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some formulas in the mentioned table. 1.
On the inversion of ... in terms of associated Stirling numbers
, 1995
"... > n+m n m z n n! : (1a) The numbers (\Gamma1) n+m \Theta n m are also called Stirling numbers of the first kind [8]. Stirling partition numbers \Phi n m \Psi , also called Stirling numbers of the second kind, are defined by (e z \Gamma 1) m = m! X n ae n m oe z n n! ; (1 ..."
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> n+m n m z n n! : (1a) The numbers (\Gamma1) n+m \Theta n m are also called Stirling numbers of the first kind [8]. Stirling partition numbers \Phi n m \Psi , also called Stirling numbers of the second kind, are defined by (e z \Gamma 1) m = m! X n ae n m oe z n n! ; (1b) and 2associated Stirling partition numbers \Phi n m \Psi 2 are defined by [2, exercise 5.7; 7, p. 296; 9, x4.5] (e z \Gamma 1 \Gamma z) m = m! X n ae n m
On the Theory of the Γq Function
, 2009
"... We consider the Γq function for 0 < q  < 1 and complex function values. qAnalogues of Euler’s constant, the Gaussian Ψ function, the Euler and Weierstrass formulas for Γ(z) are introduced. The meromorphic continuation of the Γqfunction is found. For the qRiemann zeta function [26], we show a mul ..."
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We consider the Γq function for 0 < q  < 1 and complex function values. qAnalogues of Euler’s constant, the Gaussian Ψ function, the Euler and Weierstrass formulas for Γ(z) are introduced. The meromorphic continuation of the Γqfunction is found. For the qRiemann zeta function [26], we show a multiplication formula with the Γq function. The Jacobi elliptic functions sn u, cn u and dn u may be expressed in the form sin x, cosx and 1 times a balanced Γq function. We give a solution of the Truesdell [40] Fq equation.
arXiv:1105.6063 [mathph] Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
, 2011
"... The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denomina ..."
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The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.
ON A RECIPROCITY LAW FOR FINITE MULTIPLE ZETA VALUES
"... ABSTRACT. It was shown in [11, 13] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [11, 13] can be generalized to finite variants of multiple ze ..."
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ABSTRACT. It was shown in [11, 13] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [11, 13] can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.
unknown title
, 1995
"... The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can be o ..."
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The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can be obtained for Φα in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.
GENERATORS FOR VECTOR SPACES CONSISTING OF DOUBLE ZETA VALUES WITH EVEN WEIGHT
, 802
"... Abstract. Let DZk be the Qvector space consisting of double zeta values with weight k, and DMk be its quotient space divided by the space consisting of the zeta value ζ(k) and two products of zeta values with total weight k. When k is even, an upper bound for the dimension of DMk, which gives that ..."
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Abstract. Let DZk be the Qvector space consisting of double zeta values with weight k, and DMk be its quotient space divided by the space consisting of the zeta value ζ(k) and two products of zeta values with total weight k. When k is even, an upper bound for the dimension of DMk, which gives that of DZk, is known. In this note, we obtain some sets of specific generators for DMk which represent the upper bound. These yield the corresponding sets and the upper bound for DZk. 1. Introduction and
ON A RECIPROCITY LAW FOR FINITE MULTIPLE ZETA VALUES MARKUS KUBA AND HELMUT PRODINGER
, 905
"... Abstract. It was shown in [7, 9] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [7, 9] can be generalized to finite variants of multiple zeta v ..."
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Abstract. It was shown in [7, 9] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from [7, 9] can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a simple elementary proof of the shuffle identity using only partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.
Quantization of the Optical Phase Space S 2 = {ϕ mod 2π, I> 0} in Terms of the Group SO ↑ (1, 2)
, 2003
"... The problem of quantizing properly the canonical pair “angle and action variables ”, ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantizat ..."
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The problem of quantizing properly the canonical pair “angle and action variables ”, ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1, 2): The crucial point is that the phase space S 2 = {ϕ mod 2π, I> 0} has the global structure S 1 × R + (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1, 2) acts transitively, effectively and Hamiltonlike on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space S 2 has the conic structure of an orbifold R 2 /Z2. That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2fold covering of SO(1, 2), the symplectic group Sp(2, R). The basic variables on the phase space are the 1
NNLO coefficient functions of Higgs and Drell–Yan cross sections in
, 2004
"... We calculate the Mellin moments of nexttonexttoleading order coefficient functions of the DrellYan and Higgs production cross sections. The results can be expressed in term of finite harmonic sums which are maximally threefold up to weight four. Various algebraic relations among these finite su ..."
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We calculate the Mellin moments of nexttonexttoleading order coefficient functions of the DrellYan and Higgs production cross sections. The results can be expressed in term of finite harmonic sums which are maximally threefold up to weight four. Various algebraic relations among these finite sums reduce the complexity of the results suitable for fast numerical evaluations. It is shown that only five non–trivial functions occur besides Euler’s ψ–function in the representation of these Wilson coefficients. 1.