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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
2: 25.6 Integer Arguments
§25.6(iii) Recursion Formulas
3: Bibliography W
  • J. Wimp (1968) Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.
  • 4: Bibliography F
  • G. Freud (1976) On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.
  • 5: 27.14 Unrestricted Partitions
    Multiplying the power series for f ( x ) with that for 1 / f ( x ) and equating coefficients, we obtain the recursion formula
    6: 29.20 Methods of Computation
    Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …
    7: Bibliography G
  • W. Gautschi (1961) Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.
  • W. Gautschi (1999) A note on the recursive calculation of incomplete gamma functions. ACM Trans. Math. Software 25 (1), pp. 101–107.
  • A. Gil, J. Segura, and N. M. Temme (2006c) The ABC of hyper recursions. J. Comput. Appl. Math. 190 (1-2), pp. 270–286.
  • A. Gil, J. Segura, and N. M. Temme (2007b) Numerically satisfactory solutions of hypergeometric recursions. Math. Comp. 76 (259), pp. 1449–1468.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 8: 3.5 Quadrature
    These can be found by means of the recursion
    Gauss–Laguerre Formula
    The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … a complex Gauss quadrature formula is available. …
    9: 9.19 Approximations
  • Martín et al. (1992) provides two simple formulas for approximating Ai ( x ) to graphical accuracy, one for - < x 0 , the other for 0 x < .

  • Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of Ai ( z ) , Ai ( z ) stored at the nodes. Ai ( z ) and Ai ( z ) are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of Ai ( z ) , Ai ( z ) at the node. Similarly for Bi ( z ) , Bi ( z ) .

  • 10: 12.14 The Function W ( a , x )
    §12.14(iv) Connection Formula
    12.14.9 w 1 ( a , x ) = n = 0 α n ( a ) x 2 n ( 2 n ) ! ,
    12.14.10 w 2 ( a , x ) = n = 0 β n ( a ) x 2 n + 1 ( 2 n + 1 ) ! ,
    where α n ( a ) and β n ( a ) satisfy the recursion relations …