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11—20 of 616 matching pages
11: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
.
Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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Askey-Wilson Polynomials for Root Systems of Type
.
In Hypergeometric Functions on Domains of Positivity, Jack
Polynomials, and Applications (Tampa, FL, 1991),
Contemp. Math., Vol. 138, pp. 189–204.
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Bessel Functions and their Applications.
Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.
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12: 19.37 Tables
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Functions and
… ►Functions and
►Tabulated for , to 10D by Fettis and Caslin (1964). ►Tabulated for , to 7S by Beli͡akov et al. (1962). … ►Tabulated for , , to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …13: 34.14 Tables
§34.14 Tables
►Tables of exact values of the squares of the and symbols in which all parameters are are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of , and symbols on pp. … ►Some selected symbols are also given. … 16-17; for symbols on p. … ► 310–332, and for the symbols on pp. …14: 9.4 Maclaurin Series
15: 26.5 Lattice Paths: Catalan Numbers
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is the Catalan number.
…(Sixty-six equivalent definitions of are given in Stanley (1999, pp. 219–229).)
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26.5.3
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26.5.4
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26.5.7
16: 19.36 Methods of Computation
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►If (19.36.1) is used instead of its first five terms, then the factor in Carlson (1995, (2.2)) is changed to .
►For both and the factor in Carlson (1995, (2.18)) is changed to when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms:
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►The step from to is an ascending Landen transformation if (leading ultimately to a hyperbolic case of ) or a descending Gauss transformation if (leading to a circular case of ).
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►Here is computed either by the duplication algorithm in Carlson (1995) or via (19.2.19).
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►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both and .
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17: 24.16 Generalizations
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►For , Bernoulli and Euler polynomials of order
are defined respectively by
…When they reduce to the Bernoulli and Euler numbers of
order
:
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►For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b) and López and Temme (1999b, 2010b).
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is a polynomial in of degree .
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►Generalized Bernoulli numbers and polynomials belonging to are defined by
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18: 24.9 Inequalities
19: 8.23 Statistical Applications
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►The function and its normalization play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275).
In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
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