relation%20to%20q-hypergeometric%20functions
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21: 16.13 Appell Functions
§16.13 Appell Functions
►The following four functions of two real or complex variables and cannot be expressed as a product of two functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►
16.13.1
,
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16.13.4
.
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22: 11.9 Lommel Functions
§11.9 Lommel Functions
… ► … ►§11.9(ii) Expansions in Series of Bessel Functions
… ►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … ►23: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When , (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of . …24: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
… ►It is a meromorphic function with no zeros, and with simple poles of residue at . … ►
5.2.2
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5.2.3
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25: 31.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main functions treated in this chapter are , , , and the polynomial .
…Sometimes the parameters are suppressed.
, | real variables. |
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… |
26: 12.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
…
►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
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►These notations are due to Miller (1952, 1955).
An older notation, due to Whittaker (1902), for is .
The notations are related by .
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27: 11.10 Anger–Weber Functions
§11.10 Anger–Weber Functions
… ►§11.10(vi) Relations to Other Functions
… ► … ►where the prime on the second summation symbols means that the first term is to be halved. ►§11.10(ix) Recurrence Relations and Derivatives
…28: 25.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main function treated in this chapter is the Riemann zeta function
.
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►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet -functions
.
nonnegative integers. | |
… | |
primes | on function symbols: derivatives with respect to argument. |
29: 18.27 -Hahn Class
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