recursion formulas

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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 $p\left(n\right)$ for . … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

2: 25.6 Integer Arguments

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§25.6(iii) Recursion Formulas

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3: Bibliography W

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:

FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

Encodings:

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4: Bibliography F

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

BibTeX

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5: 27.14 Unrestricted Partitions

… ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ 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

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§7.22(iii).

Encodings:

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

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A note on the recursive calculation of incomplete gamma functions. ACM Trans. Math. Software 25 (1), pp. 101–107.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

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The ABC of hyper recursions. J. Comput. Appl. Math. 190 (1-2), pp. 270–286.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

Encodings:

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Numerically satisfactory solutions of hypergeometric recursions. Math. Comp. 76 (259), pp. 1449–1468.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

Encodings:

BibTeX

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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: 18.2 General Orthogonal Polynomials

… ►The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms. … ►

§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). … ►

Degree lowering and raising differentiation formulas and structure relations

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10: 9.19 Approximations

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Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for , the other for .

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Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ 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 $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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