A Method for q-Calculus

The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy S q =The theory which emerges is usually referred to as nonextensive statistical mechanics and recovers the standard theory for q = 1. During the last two decades, this q-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture[2] and numerical indications available in the literature have been, for a few years, suggesting the possibility of q-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case q = 1 corresponding to standard probabilistic independence. This is what we prove in the present paper for 1 ≤ q < 3. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form p(x) = C q [1 − (1 − q)βx 2 ] 1/(1−q) with β > 0, and normalizing constant C q . These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional S q (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions. Mathematics Subject Classification (2000). Primary 60F05; Secondary 60E07, 60E10, 82Cxx.

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“…These generalizations were introduced by Euler [14], Jackson [15], and others. See [16] for details.…”

Section: Q-exponential and Q-logarithmmentioning

confidence: 99%

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

2008

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“…The nominal numerical methods for approximating integrals do not seem to be appropriate for q-integrals. We could find less works for developing numerical procedures for accurate numerical solutions [9]- [14]. In this paper, we present a procedure for the numerical q-calculation of the q-integrals based on appropriate nodes and weights which are determined such that the error of q-integration is minimized.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights

Hashemiparast¹,

Ghondaghsaz²,

Maghasedi³

2015

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In this paper, we present a procedure for the numerical q-calculation of the q-integrals based on appropriate nodes and weights which are determined such that the error of q-integration is minimized; a system of linear and nonlinear set of equations respectively are prepared to obtain the nodes and weights simultaneously; the error of q-integration is considered to be minimized under this condition; finally some application and numerical examples are given for comparison with the exact solution. At the end, the related tables of approximations are presented.

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“…We will now describe the q-umbral method invented by the author [18][19][20][21][22][23][24], which also involves the Nalli-Ward-AlSalam (NWA) q-addition and the Jackson-Hahn-Cigler (JHC) q-addition. This method is a mixture of Heine [36] and Gasper and Rahman [31].…”

Section: Introductionmentioning

confidence: 99%

“…In [14] a parameter augmentation method for a reciprocal of a q-shifted factorial was used to obtain q-summation formulas. In [20] the equivalent approach (by the q-binomial theorem) to use the q-exponential function E q to obtain formulas for q-Laguerre polynomials was used. As Fujiwara [30] showed, the most important property of the Jacobi, Laguerre, and Hermite polynomials is the generalized Rodriguez formula.…”

Section: Introductionmentioning

confidence: 99%

q-Calculus as operational algebra

Ernst¹

2009

Proc. Estonian Acad. Sci.

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Abstract. This second paper on operational calculus is a continuation of Ernst, T. q-Analogues of some operational formulas. Algebras Groups Geom., 2006, 23(4) Padova, 1965, 35, 237-244). A field of fractions for Cigler's multiplication operator (Cigler, J. Operatormethoden für q-Identitäten II, q-Laguerre-Polynome. Monatsh. Math., 1981, 91, 105-117) is used in the computations. The formulas for q-Jacobi polynomials are mostly formal. We find q-orthogonality relations for q-Laguerre, q-Jacobi, and q-Legendre polynomials using q-integration by parts. This q-Legendre polynomial is given here for the first time, we also find its q-difference equations.

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A Method for q-Calculus

Cited by 114 publications

References 59 publications

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights

q-Calculus as operational algebra

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