where A and B are two formal power series subject to the conditions A(0) B(k)(0) = 0, k = 0, 1, 2, .... In this work, we shall determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sequences...
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where A and B are two formal power series subject to the conditions A(0) B(k)(0) = 0, k = 0, 1, 2, .... In this work, we shall determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sequences, that is to say, polynomials with Brenke type generating function and satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of d-orthogonality. The classification is based on the discussion of a three-order difference equation induced by the fourterm recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara (1968) who gave all pairs (A(t), B(t)) for which {Pn(x)}n >= 0 is an orthogonal polynomial sequence. In some cases, we give the expression of the moments associated to the two-dimensional functional of orthogonality. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let N (sic) G be a pair of finite subgroups of SL2(C) and V a finite-dimensional fundamental G-module. We study Kostant's generating functions for the decomposition of the SL2(C)-module S-k(V) restricted to N (sic...
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Let N (sic) G be a pair of finite subgroups of SL2(C) and V a finite-dimensional fundamental G-module. We study Kostant's generating functions for the decomposition of the SL2(C)-module S-k(V) restricted to N (sic) G in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincare series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.
MacMahon's Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. ...
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MacMahon's Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. We introduce a new generalized generating function that gives, as special cases, extensions of the generating functions of the Banzhaf, Shapley-Shubik, Banzhaf-Owen, symmetric coalitional Banzhaf, and Owen power indices. Our extensions involve any coalition formation related to a linear diophantine system and multiple voting games. In addition, we show that a combination of ideas from MPA and Clifford algebras is useful in constructing generating functions for coalition configuration power indices. Finally, a brief account on how to design voting systems via MPA is advanced. More precisely, we obtain new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not.
In this paper, some generating functions involving 3-variable Hermite-Laguerre polynomials are derived by using operational methods. Further, generating functions for the polynomials related to Hermite-Laguerre polyno...
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In this paper, some generating functions involving 3-variable Hermite-Laguerre polynomials are derived by using operational methods. Further, generating functions for the polynomials related to Hermite-Laguerre polynomials are also obtained as applications of the main results. (C) 2014 Elsevier Inc. All rights reserved.
In a recent paper, Srivastava et al. (2012) [23] introduced and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. The main object of this pa...
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In a recent paper, Srivastava et al. (2012) [23] introduced and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. The main object of this paper is to investigate several generating functions for a certain class of incomplete hypergeometric polynomials associated with them. Various (known or new) special cases and consequences of the results presented in this paper are also considered. (C) 2012 Elsevier Inc. All rights reserved.
We consider a cooperative game based on a network in which nodes represent players and the characteristic function is defined using a maximal covering by the pairs of connected nodes. Problems of this form arise in ma...
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We consider a cooperative game based on a network in which nodes represent players and the characteristic function is defined using a maximal covering by the pairs of connected nodes. Problems of this form arise in many applications such as mobile communications, patrolling, logistics and sociology. The Owen value, which describes the significance of each node in the network, is derived. We show that the method of generating functions can be useful for calculating this Owen value and illustrate this approach based on examples of network structures. (C) 2020 Elsevier B.V. All rights reserved.
We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio q, we c...
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We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio q, we construct a PDE for these generating functions in which classical derivatives are replaced by q-derivatives. We then solve this PDE for the generating functions using q-exponential functions. We apply our generating functions to derive some known and some novel identities for B-splines with knots in geometric or affine progression, including a generalization of the Schoenberg identity, formulas for sums and alternating sums, and an explicit expression for the moments of these B-splines. Special cases include both the uniform B-splines with knots at the integers and the nonuniform B-splines with knots at the q-integers.
We investigate the arithmetic-geometric structure of the lecture hall cone L-n :- {lambda is an element of R-n : 0 <= lambda(1)/1 <= lambda(2)/2 <= lambda(3)/3 <= ... <= lambda(n/)n} We show that L-n is...
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We investigate the arithmetic-geometric structure of the lecture hall cone L-n :- {lambda is an element of R-n : 0 <= lambda(1)/1 <= lambda(2)/2 <= lambda(3)/3 <= ... <= lambda(n/)n} We show that L-n is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h*-polynomial is given by the (n-1) st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for L-n, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of L-n, including connections between enumerative and algebraic properties of L-n and cones over unit cubes.
In this paper, we introduce a new operator in order to derive some new symmetric properties of k-Fibonacci and k-Pell numbers and Tchebychev polynomials of first and second kind. By making use of the new operator defi...
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In this paper, we introduce a new operator in order to derive some new symmetric properties of k-Fibonacci and k-Pell numbers and Tchebychev polynomials of first and second kind. By making use of the new operator defined in this paper, we give some new generating functions for k-Fibonacci and k-Pell numbers and Fibonacci polynomials.
We derive explicit formulas for the generating functions of the q-Bernstein basis functions in terms of q-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these ...
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We derive explicit formulas for the generating functions of the q-Bernstein basis functions in terms of q-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these generating functions which we apply to prove a variety of identities, some old and some new, for the q-Bernstein bases.
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