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11: 31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values q m , m = 0 , 1 , 2 , , of the accessory parameter q , the function H ( a , q ; α , β , γ , δ ; z ) is analytic at z = 1 , and hence also throughout the disk | z | < a . … The eigenvalues q m satisfy the continued-fraction equation
31.4.2 q = a γ P 1 Q 1 + q - R 1 P 2 Q 2 + q - R 2 P 3 Q 3 + q - ,
12: 20.16 Software
§20.16(ii) Real Argument and Parameter
§20.16(iii) Complex Argument and/or Parameter
13: 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. The total number of free parameters is six. … Then (suppressing the parameter k ) … Except for the identity automorphism, each alters the parameters.
14: 8.17 Incomplete Beta Functions
Throughout §§8.17 and 8.18 we assume that a > 0 , b > 0 , and 0 x 1 . …
8.17.4 I x ( a , b ) = 1 - I 1 - x ( b , a ) .
With a > 0 , b > 0 , and 0 < x < 1 , …
8.17.12 I x ( a , b ) = x I x ( a - 1 , b ) + x I x ( a , b - 1 ) ,
8.17.13 ( a + b ) I x ( a , b ) = a I x ( a + 1 , b ) + b I x ( a , b + 1 ) ,
15: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
16: 28.27 Addition Theorems
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …
17: 31.12 Confluent Forms of Heun’s Equation
31.12.1 d 2 w d z 2 + ( γ z + δ z - 1 + ϵ ) d w d z + α z - q z ( z - 1 ) w = 0 .
31.12.2 d 2 w d z 2 + ( δ z 2 + γ z + 1 ) d w d z + α z - q z 2 w = 0 .
31.12.3 d 2 w d z 2 - ( γ z + δ + z ) d w d z + α z - q z w = 0 .
31.12.4 d 2 w d z 2 + ( γ + z ) z d w d z + ( α z - q ) w = 0 .
18: 32.1 Special Notation
m , n

integers.

k

real parameter.

19: 34.14 Tables
Tables of exact values of the squares of the 3 j and 6 j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 j , 6 j , and 9 j symbols on pp. … Tables of 3 j and 6 j symbols in which all parameters are 17 / 2 are given in Appel (1968) to 6D. … In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters 5 and numerical values for all parameters 3 are given on pp. …
20: 15.5 Derivatives and Contiguous Functions
15.5.11 ( c - a ) F ( a - 1 , b ; c ; z ) + ( 2 a - c + ( b - a ) z ) F ( a , b ; c ; z ) + a ( z - 1 ) F ( a + 1 , b ; c ; z ) = 0 ,