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Gauss–Jacobi formula

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1: 3.5 Quadrature
GaussJacobi Formula
2: 20.11 Generalizations and Analogs
§20.11(i) Gauss Sum
This is Jacobi’s inversion problem of §20.9(ii). … Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …
3: 18.12 Generating Functions
Jacobi
18.12.1 2 α + β R ( 1 + R - z ) α ( 1 + R + z ) β = n = 0 P n ( α , β ) ( x ) z n , R = 1 - 2 x z + z 2 , | z | < 1 .
18.12.3 ( 1 + z ) - α - β - 1 F 1 2 ( 1 2 ( α + β + 1 ) , 1 2 ( α + β + 2 ) β + 1 ; 2 ( x + 1 ) z ( 1 + z ) 2 ) = n = 0 ( α + β + 1 ) n ( β + 1 ) n P n ( α , β ) ( x ) z n , | z | < 1 ,
and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function F 1 2 see §§15.1, 15.2(i). …
4: Errata
  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Chapter 35 Functions of Matrix Argument

    The generalized hypergeometric function of matrix argument F q p ( a 1 , , a p ; b 1 , , b q ; T ) , was linked inadvertently as its single variable counterpart F q p ( a 1 , , a p ; b 1 , , b q ; T ) . Furthermore, the Jacobi function of matrix argument P ν ( γ , δ ) ( T ) , and the Laguerre function of matrix argument L ν ( γ ) ( T ) , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by P ν ( γ , δ ) ( T ) , and L ν ( γ ) ( T ) . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

  • Subsections 15.4(i), 15.4(ii)

    Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula F ( a , b ; a ; z ) = ( 1 - z ) - b .

    Report by Louis Klauder on 2017-01-01.

  • Table 18.3.1

    Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

  • 5: 18.5 Explicit Representations
    §18.5(ii) Rodrigues Formulas
    Related formula: …
    Jacobi
    and two similar formulas by symmetry; compare the second row in Table 18.6.1. … For corresponding formulas for Chebyshev, Legendre, and the Hermite He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
    6: Bibliography M
  • T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
  • S. C. Milne (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • 7: 31.7 Relations to Other Functions
    §31.7(i) Reductions to the Gauss Hypergeometric Function
    Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . …
    31.7.2 H ( 2 , α β ; α , β , γ , α + β - 2 γ + 1 ; z ) = F 1 2 ( 1 2 α , 1 2 β ; γ ; 1 - ( 1 - z ) 2 ) ,
    With z = sn 2 ( ζ , k ) and …Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities ζ = K , K + i K , and i K , where K and K are related to k as in §19.2(ii).
    8: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 9: 15.9 Relations to Other Functions
    Jacobi
    §15.9(ii) Jacobi Function
    The Jacobi transform is defined as …with inverse … …
    10: Bibliography I
  • M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.
  • M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and q -Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
  • A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
  • A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.