beta function

(redirected from Beta integral)

beta function

[′bād·ə ‚fənk·shən]
(mathematics)
A function of two positive variables, defined by
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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We will then identify a natural setting, namely that of q-generalizations of the Dixon Anderson and Selberg integrals both multidimensional generalizations of the Euler beta integral as candidates for permitting analysis from this perspective and as providing new multi-dimensional extensions of (1).
In the limit q [right arrow] 1 this reduces to the Euler beta integral, while taking q [right arrow] 1 in the recurrence (6) implies the beta integral evaluation
Ramanujan's [sub.1][psi.sub.1] summation theorem--perspective, announcement of bilateral q-Dixon-Anderson and q-Selberg integral extensions, and context--
Among the topics are orthogonal polynomials spanning a non-standard flag, the supersymmetric spectra of two planar integrable quantum systems, solvable rational extension of translationally shaped invariants potentials, explicit higher-dimension Darboux transformations for the time-dependent Schrodinger equation, and elliptic beta integrals and solvable models of statistical mechanics.
Algebraic aspects of Darboux transformations, quantum integrable systems, and supersymmetric quantum mechanics; proceedings
Next, we transform the [t.sub.i]-integrals to well known multiple Beta integrals by the following transformation:
A theorem connecting the H-transform and fractional integral operators involving the multivariable H-function

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