Jacobi transform

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31: 18.38 Mathematical Applications

… ►It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►The Askey–Gasper inequality … ►

Radon Transform

… ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

32: Bibliography E

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

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Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

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Tables of Integral Transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Table errata: Math. Comp. v. 66 (1997), no. 220, p. 1766–1767, v. 65 (1996), no. 215, p. 1384, v. 50 (1988), no. 182, p. 653, v. 41 (1983), no. 164, p. 778–779, v. 27 (1973), no. 122, p. 451, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239-240.

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MathReview (H. Kober), ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.10(iii), §13.10(iv), §13.23(i), §13.23(ii), §14.17(v), §15.14, §16.20, §16.5, (18.17.21_1), (18.17.34_5), §18.17(ix), §20.10(ii), §20.10(iii), §25.5(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.14.

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Tables of Integral Transforms. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, pp. 779–780, v. 31 (1977), no. 138, p. 614, v. 31 (1977), no. 137, pp. 328–329, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 23 (1969), no. 106, p. 468.

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MathReview (H. Kober), ZentralBlatt

Referenced by:

§1.14(viii), §10.22(v), §10.22(vi), §10.43(v), §10.43(vi), §12.12, §13.10(v), §13.10(v), §13.10(vi), §13.23(iii), §13.23(v), §15.14, §15.14, §18.17(ix), §5.13, §8.14.

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On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

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Notes:

With a loose Russian summary

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ISSN 0011-4642, MathReview (D. Hinton)

Referenced by:

§1.13(viii).

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33: 20.11 Generalizations and Analogs

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. … ►For $m=1,2,3,4$, $n=1,2,3,4$, and $m\ne n$, define twelve combined theta functions ${\phi }_{m,n}(z,q)$ by …

34: 18.2 General Orthogonal Polynomials

… ►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. … … ►

§18.2(vi) Zeros

… ►When the Jacobi matrix in (18.2.11_9) is truncated to an $n\times n$ matrix … ►

§18.2(vii) Quadratic Transformations

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35: 31.2 Differential Equations

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Jacobi’s Elliptic Form

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$F$-Homotopic Transformations

… ►By composing these three steps, there result $2^3=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►

Homographic Transformations

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Composite Transformations

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36: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

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Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

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Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (43), pp. 43016, 16 pp..

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ISSN 1751-8113, Document, Link, MathReview (María-José Cantero)

Referenced by:

§18.36(vi).

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

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37: Bibliography L

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

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ISSN 0010-4655, Document, Code, ZentralBlatt

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4th item.

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Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

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Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

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Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

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FullText, MathReview (I. A. Stegun), ZentralBlatt

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2nd item.

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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38: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. … ►Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. …

39: Bibliography J

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Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

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Notes:

Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.

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Werke TOC

Referenced by:

§27.13(iv).

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

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Fast Hankel transforms. Geophysical Prospecting 27 (4), pp. 876–901.

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Document

Referenced by:

§10.74(vii).

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

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