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1: 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. If the other exponent is not a positive integer, that is, if γ 0 , - 1 , - 2 , , then from §2.7(i) it follows that H ( a , q ; α , β , γ , δ ; z ) exists, is analytic in the disk | z | < 1 , and has the Maclaurin expansion … Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. …For example, H ( a , q ; α , β , γ , δ ; z ) is equal to … The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
2: 4.42 Solution of Triangles
3: 31.7 Relations to Other Functions
31.7.1 F 1 2 ( α , β ; γ ; z ) = H ( 1 , α β ; α , β , γ , δ ; z ) = H ( 0 , 0 ; α , β , γ , α + β + 1 - γ ; z ) = H ( a , a α β ; α , β , γ , α + β + 1 - γ ; z ) .
Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . …
31.7.2 H ( 2 , α β ; α , β , γ , α + β - 2 γ + 1 ; z ) = F 1 2 ( 1 2 α , 1 2 β ; γ ; 1 - ( 1 - z ) 2 ) ,
31.7.3 H ( 4 , α β ; α , β , 1 2 , 2 3 ( α + β ) ; z ) = F 1 2 ( 1 3 α , 1 3 β ; 1 2 ; 1 - ( 1 - z ) 2 ( 1 - 1 4 z ) ) ,
31.7.4 H ( 1 2 + i 3 2 , α β ( 1 2 + i 3 6 ) ; α , β , 1 3 ( α + β + 1 ) , 1 3 ( α + β + 1 ) ; z ) = F 1 2 ( 1 3 α , 1 3 β ; 1 3 ( α + β + 1 ) ; 1 - ( 1 - ( 3 2 - i 3 2 ) z ) 3 ) .
4: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) = H ( a , q n , m ; - n , β , γ , δ ; z )
5: 31.1 Special Notation
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) . …
6: 31.9 Orthogonality
31.9.3 θ m = ( 1 - e 2 π i γ ) ( 1 - e 2 π i δ ) ζ γ ( 1 - ζ ) δ ( ζ - a ) ϵ f 0 ( q , ζ ) f 1 ( q , ζ ) q 𝒲 { f 0 ( q , ζ ) , f 1 ( q , ζ ) } | q = q m ,
f 0 ( q m , z ) = H ( a , q m ; α , β , γ , δ ; z ) ,
f 1 ( q m , z ) = H ( 1 - a , α β - q m ; α , β , δ , γ ; 1 - z ) ,
31.9.6 ρ ( s , t ) = ( s - t ) ( s t ) γ - 1 ( ( s - 1 ) ( t - 1 ) ) δ - 1 ( ( s - a ) ( t - a ) ) ϵ - 1 ,
7: 2.6 Distributional Methods
Let f ( t ) be locally integrable on [ 0 , ) . The Stieltjes transform of f ( t ) is defined by …Since f ( t ) is locally integrable on [ 0 , ) , it defines a distribution by … In terms of the convolution product …of two locally integrable functions on [ 0 , ) , (2.6.33) can be written …
8: 28.19 Expansions in Series of me ν + 2 n Functions
28.19.2 f ( z ) = n = - f n me ν + 2 n ( z , q ) ,
28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( - z , q ) d z .
9: 33.11 Asymptotic Expansions for Large ρ
10: 18.14 Inequalities
§18.14(iii) Local Maxima and Minima
Jacobi
Laguerre