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Jackson–Bessel Function
In mathematics, a Jackson ''q''-Bessel function (or basic Bessel function) is one of the three ''q''-analogs of the Bessel function introduced by . The third Jackson ''q''-Bessel function is the same as the Hahn–Exton ''q''-Bessel function. Definition The three Jackson ''q''-Bessel functions are given in terms of the ''q''-Pochhammer symbol and the basic hypergeometric function \phi by : J_\nu^(x;q) = \frac (x/2)^\nu _2\phi_1(0,0;q^;q,-x^2/4), \quad , x, -1, the second Jackson ''q''-Bessel function satisfies: \left, J_^(z;q)\\leq\frac\left(\frac\right)^\nu\exp\left\. (see .) For n\in\mathbb, \left, J_^(z;q)\\leq\frac\left(\frac\right)^n(-, z, ^2;q)_. (see .) Generating Function The following formulas are the ''q''-analog of the generating function for the Bessel function (see ): :\sum_^t^nJ_n^(x;q)=(-x^2/4;q)_e_q(xt/2)e_q(-x/2t), :\sum_^t^nJ_n^(x;q)=e_q(xt/2)E_q(-qx/2t). e_q is the ''q''-exponential function. Alternative Representations Integral Representations The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn–Exton Q-Bessel Function
In mathematics, the Hahn–Exton ''q''-Bessel function or the third Jackson q-Bessel function, Jackson ''q''-Bessel function is a q-analog, ''q''-analog of the Bessel function, and satisfies the Hahn-Exton ''q''-difference equation (). This function was introduced by in a special case and by in general. The Hahn–Exton ''q''-Bessel function is given by : J_\nu^(x;q) = \frac \sum_\frac= \frac x^\nu _1\phi_1(0;q^;q,qx^2). \phi is the basic hypergeometric function. Properties Zeros Koelink and Swarttouw proved that J_\nu^(x;q) has infinite number of real zeros. They also proved that for \nu>-1 all non-zero roots of J_\nu^(x;q) are real (). For more details, see and . Zeros of the Hahn-Exton ''q''-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (, ) Derivatives For the (usual) derivative and ''q''-derivative of J_\nu^(x;q), see . The symmetric ''q''-derivative of J_\nu^(x;q) is described on . Recurrence R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Pochhammer Symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Hypergeometric Function
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Monotonic Function
In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions. Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him. There are several other related notions like the concepts of almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-exponential
In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a ''q''-derivative. There are many ''q''-derivatives, for example, the classical ''q''-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, ''q''-exponentials are not unique. For example, e_q(z) is the ''q''-exponential corresponding to the classical ''q''-derivative while \mathcal_q(z) are eigenfunctions of the Askey-Wilson operators. Definition The ''q''-exponential e_q(z) is defined as :e_q(z)= \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac where _q is the ''q''-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the ''q''-Pochhammer symbol. That this is the ''q''-analog of the exponential follows from the property :\left(\frac\right)_q e_q(z) = e_q(z) where the derivative on the left is the ''q''-derivative. The above is easily verified by considering the ''q''-derivative of the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Bessel Polynomials
In mathematics, the ''q''-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s by Roelof Koekoek, Peter Lesky Rene Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010： :y_(x;a;q)=\;_2\phi_1 \left(\begin q^ & -aq^ \\ 0 \end ; q,qx \right). Also known as alternative q-Charlier polynomials K(x;a;q). Orthogonality : \sum_^\left(\frac*q^*y_*(q^k;a;q)*y_*(q^k;a;q)\right)=(q;q)_*(-aq^n;q)_\frac\delta_ Roelof p527 where (q;q)_n\text(-aq^n;q)_\infty are q-Pochhammer symbols. Gallery References * * *{{dlmf, id=1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Nachrichten
''Mathematische Nachrichten'' (abbreviated ''Math. Nachr.''; English: ''Mathematical News'') is a mathematical journal published in 12 issues per year by Wiley-VCH GmbH. It should not be confused with the ''Internationale Mathematische Nachrichten'', an unrelated publication of the Austrian Mathematical Society. It was established in 1948 by East German mathematician Erhard Schmidt, who became its first editor-in-chief. At that time it was associated with the German Academy of Sciences at Berlin, and published by Akademie Verlag. After the fall of the Berlin Wall, Akademie Verlag was sold to VCH Verlagsgruppe Weinheim, which in turn was sold to John Wiley & Sons. According to the 2020 edition of Journal Citation Reports, the journal had an impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Analysis And Applications
The ''Journal of Mathematical Analysis and Applications'' is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960, as part of a series of new journals on areas of mathematics published by Academic Press, and is now published by Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', .... For most years since 1997 it has been ranked by SCImago Journal Rank as among the top 50% of journals in its topic areas. retrieved 201 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
