relation to power series

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1: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

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2: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. …

3: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

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4: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►If $n=2$, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … ►Then $T_N$ has at most one term if $N\le 5$ in the series for $R_F$. … ►

5: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E1

Encodings:

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See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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6: 13.14 Definitions and Basic Properties

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Whittaker’s Equation

… ►Standard solutions are: … ►The series … ►

§13.14(iii) Limiting Forms as $z\to 0$

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§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

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7: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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8: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

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§18.26(ii) Limit Relations

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Wilson $\to$ Jacobi

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§18.26(iii) Difference Relations

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§18.26(iv) Generating Functions

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9: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …

10: 24.17 Mathematical Applications

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24.17.7 $$M_n(x)=O\left({|x|}^{\gamma }\right),$$ $x\to \pm \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $n$: integer, $x$: real or complex and $M_n(x)$: function

Permalink:

http://dlmf.nist.gov/24.17.E7

Encodings:

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See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …