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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: 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 ) .
3: 4.42 Solution of Triangles
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: 37.20 Mathematical Applications
For regular domains, such as square, sphere, ball, simplex, and conic domains, they are used to study convolution structure, maximal functions, and interpolation spaces, as well as localized kernel and localized frames. …
8: 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 …
9: 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 .
10: 33.11 Asymptotic Expansions for Large ρ