orthogonal functions with respect to weighted summation
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1: 9.1 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 the Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to
and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii). | |
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primes | derivatives with respect to argument. |
2: 23.15 Definitions
§23.15 Definitions
►§23.15(i) General Modular Functions
… ►Elliptic Modular Function
… ►Dedekind’s Eta Function (or Dedekind Modular Function)
… ►3: 5.15 Polygamma Functions
§5.15 Polygamma Functions
►The functions , , are called the polygamma functions. In particular, is the trigamma function; , , are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For see §24.2(i). …4: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . ►§20.2(ii) Periodicity and Quasi-Periodicity
… ►The theta functions are quasi-periodic on the lattice: … ►§20.2(iv) -Zeros
…5: 11.9 Lommel Functions
§11.9 Lommel Functions
… ► ►Reflection Formulas
… ►§11.9(ii) Expansions in Series of Bessel Functions
… ►6: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
►The hypergeometric function is defined by the Gauss series … … ►§15.2(ii) Analytic Properties
… ►Because of the analytic properties with respect to , , and , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …7: 9.12 Scorer Functions
8: 14.20 Conical (or Mehler) Functions
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