SL%282%2CZ%29 bilinear transformation
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1: 1.14 Integral Transforms
§1.14 Integral Transforms
►§1.14(i) Fourier Transform
… ►§1.14(iii) Laplace Transform
… ►Fourier Transform
… ►Laplace Transform
…2: 23.15 Definitions
…
►Also denotes a bilinear transformation on , given by
…The set of all bilinear transformations of this form is denoted by SL
(Serre (1973, p. 77)).
►A modular function
is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL
,
…(Some references refer to as the level).
…
3: 23.18 Modular Transformations
§23.18 Modular Transformations
… ►and is a cusp form of level zero for the corresponding subgroup of SL . … ► is a modular form of level zero for SL . … ►
23.18.5
…
►Note that is of level .
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4: 15.17 Mathematical Applications
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►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations.
…
►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL
, and spherical functions on certain nonsymmetric Gelfand pairs.
Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform.
…
►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
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5: Bibliography F
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►
Tablicy značeniĭ funkcii ot kompleksnogo argumenta.
Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un.
J. Funct. Anal. 49 (2), pp. 230–268.
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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions.
Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
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Application of the -function theory of Painlevé equations to random matrices: , the JUE, CyUE, cJUE and scaled limits.
Nagoya Math. J. 174, pp. 29–114.
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►
On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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6: 26.12 Plane Partitions
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►The number of self-complementary plane partitions in is
…in it is
…in it is
…
►The number of symmetric self-complementary plane partitions in is
…in it is
…
7: 8.6 Integral Representations
…
►In (8.6.10)–(8.6.12), is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at , in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at .
►
8.6.10
, ,
►
8.6.11
,
►
8.6.12
, .
…
►For collections of integral representations of and see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).
8: Bibliography L
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Eine Verallgemeinerung der Sphäroidfunktionen.
Arch. Math. 11, pp. 29–39.
…
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Lie algebraic approaches to classical partition identities.
Adv. in Math. 29 (1), pp. 15–59.
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Orthogonal polynomials for exponential weights on
.
J. Approx. Theory 134 (2), pp. 199–256.
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On the function
.
J. Math. Phys. Mass. Inst. Tech. 21, pp. 264–283.
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Evaluating infinite integrals involving products of Bessel functions of arbitrary order.
J. Comput. Appl. Math. 64 (3), pp. 269–282.
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9: Bibliography G
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A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function.
J. Comput. Phys. 42 (2), pp. 277–287.
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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane.
Math. Comp. 21 (97), pp. 18–29.
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Algorithm 236: Bessel functions of the first kind.
Comm. ACM 7 (8), pp. 479–480.
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A table of partitions.
Proc. London Math. Soc. (2) 39, pp. 142–149.
►
A table of partitions (II).
Proc. London Math. Soc. (2) 42, pp. 546–549.
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