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1: 32.17 Methods of Computation
For numerical studies of P II  see Rosales (1978), Miles (1978, 1980), Kashevarov (1998, 2004), and S.  Olver (2011). …
2: 15.1 Special Notation
3: 14.21 Definitions and Basic Properties
14.21.1 ( 1 - z 2 ) d 2 w d z 2 - 2 z d w d z + ( ν ( ν + 1 ) - μ 2 1 - z 2 ) w = 0 .
4: Frank W. J. Olver
Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olvers 65th birthday, with Proceedings published by Marcel Dekker in 1990. … Fitting capstones to Olvers long career, these works are on track to extend the legacy of the classic Abramowitz and Stegun handbook well into the 21st century. According to DLMF Project Lead Daniel Lozier, “Olvers encyclopedic knowledge of the field, his clear vision for mathematical exposition, his keen sense of the needs of practitioners, and his unfailing attention to detail were key to the success of that project. …
5: 13.1 Special Notation
The main functions treated in this chapter are the Kummer functions M ( a , b , z ) and U ( a , b , z ) , Olvers function M ( a , b , z ) , and the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) . …
6: 14.3 Definitions and Hypergeometric Representations
is Olvers hypergeometric function (§15.1). …
14.3.19 Q ν μ ( x ) = 2 ν Γ ( ν + 1 ) ( x + 1 ) μ / 2 ( x - 1 ) ( μ / 2 ) + ν + 1 F ( ν + 1 , ν + μ + 1 ; 2 ν + 2 ; 2 1 - x ) ,
7: 13.10 Integrals
13.10.1 M ( a , b , z ) d z = 1 a - 1 M ( a - 1 , b - 1 , z ) ,
13.10.3 0 e - z t t b - 1 M ( a , c , k t ) d t = Γ ( b ) z - b F 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
13.10.4 0 e - z t t b - 1 M ( a , b , t ) d t = z - b ( 1 - 1 z ) - a , b > 0 , z > 1 ,
13.10.5 0 e - t t b - 1 M ( a , c , t ) d t = Γ ( b ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) , ( c - a ) > b > 0 ,
13.10.10 0 t λ - 1 M ( a , b , - t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( a ) Γ ( b - λ ) , 0 < λ < a ,
8: 15.7 Continued Fractions
15.7.1 F ( a , b ; c ; z ) F ( a , b + 1 ; c + 1 ; z ) = t 0 - u 1 z t 1 - u 2 z t 2 - u 3 z t 3 - ,
9: 15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
10: 3.6 Linear Difference Equations
§3.6(v) Olvers Algorithm
The backward recursion can be carried out using independently computed values of J N ( 1 ) and J N + 1 ( 1 ) or by use of Miller’s algorithm (§3.6(iii)) or Olvers algorithm (§3.6(v)). … Thus the asymptotic behavior of the particular solution E n ( 1 ) is intermediate to those of the complementary functions J n ( 1 ) and Y n ( 1 ) ; moreover, the conditions for Olvers algorithm are satisfied. …
Table 3.6.1: Weber function w n = E n ( 1 ) computed by Olvers algorithm.
n p n e n e n / ( p n p n + 1 ) w n