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1: 31.8 Solutions via Quadratures
For half-odd-integer values of the exponent parameters: … The curve Γ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for m j . …
2: 31.14 General Fuchsian Equation
The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N - 2 free accessory parameters q j . …
3: 31.2 Differential Equations
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, { } , can be transformed into (31.2.1). The parameters play different roles: a is the singularity parameter; α , β , γ , δ , ϵ are exponent parameters; q is the accessory parameter. …
4: 27.2 Functions
is the sum of the α th powers of the divisors of n , where the exponent α can be real or complex. …
5: 31.15 Stieltjes Polynomials
If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index m = ( m 1 , m 2 , , m N - 1 ) , where each m j is a nonnegative integer, there is a unique Stieltjes polynomial with m j zeros in the open interval ( a j , a j + 1 ) for each j = 1 , 2 , , N - 1 . …
6: 30.2 Differential Equations
30.2.1 d d z ( ( 1 - z 2 ) d w d z ) + ( λ + γ 2 ( 1 - z 2 ) - μ 2 1 - z 2 ) w = 0 .
This equation has regular singularities at z = ± 1 with exponents ± 1 2 μ and an irregular singularity of rank 1 at z = (if γ 0 ). The equation contains three real parameters λ , γ 2 , and μ . … With ζ = γ z Equation (30.2.1) changes to
30.2.4 ( ζ 2 - γ 2 ) d 2 w d ζ 2 + 2 ζ d w d ζ + ( ζ 2 - λ - γ 2 - γ 2 μ 2 ζ 2 - γ 2 ) w = 0 .
7: 31.3 Basic Solutions
H ( a , q ; α , β , γ , δ ; z ) denotes the solution of (31.2.1) that corresponds to the exponent 0 at z = 0 and assumes the value 1 there. … Similarly, if γ 1 , 2 , 3 , , then the solution of (31.2.1) that corresponds to the exponent 1 - γ at z = 0 is … Solutions of (31.2.1) corresponding to the exponents 0 and 1 - δ at z = 1 are respectively, … Solutions of (31.2.1) corresponding to the exponents 0 and 1 - ϵ at z = a are respectively, … Solutions of (31.2.1) corresponding to the exponents α and β at z = are respectively, …
8: 31.11 Expansions in Series of Hypergeometric Functions
31.11.3 λ + μ = γ + δ - 1 = α + β - ϵ .
31.11.7 L j = a ( λ + j ) ( μ - j ) - q + ( j + α - μ ) ( j + β - μ ) ( j + γ - μ ) ( j + λ ) ( 2 j + λ - μ ) ( 2 j + λ - μ + 1 ) + ( j - α + λ ) ( j - β + λ ) ( j - γ + λ ) ( j - μ ) ( 2 j + λ - μ ) ( 2 j + λ - μ - 1 ) ,
Then the Fuchs–Frobenius solution at belonging to the exponent α has the expansion (31.11.1) with … For example, consider the Heun function which is analytic at z = a and has exponent α at . …In this case the accessory parameter q is a root of the continued-fraction equation …
9: 28.29 Definitions and Basic Properties
10: 28.2 Definitions and Basic Properties
28.2.16 cos ( π ν ) = w I ( π ; a , q ) = w I ( π ; a , - q ) .