generalized hypergeometric function 0F2
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11: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
… βΊ
15.10.1
…
βΊ
Singularity
… βΊSingularity
… βΊSingularity
…12: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
βΊ§14.19(i) Introduction
… βΊ§14.19(ii) Hypergeometric Representations
βΊWith as in §14.3 and , … βΊ§14.19(v) Whipple’s Formula for Toroidal Functions
…13: 16.13 Appell Functions
§16.13 Appell Functions
βΊThe 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): βΊ
16.13.1
,
…
βΊ
16.13.4
.
…
βΊ
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14: 9.1 Special Notation
…
βΊ(For other notation see Notation for the Special Functions.)
βΊ
βΊ
βΊThe main functions treated in this chapter are the Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
βΊOther notations that have been used are as follows: and for and (Jeffreys (1928), later changed to and ); , (Fock (1945)); (SzegΕ (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii). | |
… |
15: 23.15 Definitions
§23.15 Definitions
βΊ§23.15(i) General Modular Functions
… βΊIf, as a function of , is analytic at , then is called a modular form. … βΊDedekind’s Eta Function (or Dedekind Modular Function)
… βΊ16: 14.20 Conical (or Mehler) Functions
…
βΊ
…
βΊLastly, for the range , is a real-valued solution of (14.20.1); in terms of (which are complex-valued in general):
…
βΊ
§14.20(vi) Generalized Mehler–Fock Transformation
… βΊ§14.20(viii) Asymptotic Approximations: Large ,
… βΊ§14.20(ix) Asymptotic Approximations: Large ,
…17: 11.9 Lommel Functions
§11.9 Lommel Functions
… βΊcan be regarded as a generalization of (11.2.7). Provided that , (11.9.1) has the general solution … βΊ … βΊ18: 20.2 Definitions and Periodic Properties
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βΊ