relation to confluent hypergeometric functions

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31: 13.31 Approximations

§13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. … ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). … ►

13.31.3 $$z^{a}U(a,1+a-b,z)=\underset{n\to \mathrm{\infty }}{lim}\frac{A_n(z)}{B_n(z)}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer, $z$: complex variable, $A_n(z)$: expansion and $B_n(z)$: expansion

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http://dlmf.nist.gov/13.31.E3

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Annotations for §13.31(iii), §13.31 and Ch.13

32: 13.8 Asymptotic Approximations for Large Parameters

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►To obtain approximations for $M(a,b,z)$ and $U(a,b,z)$ that hold as $b\to \mathrm{\infty }$, with $a>\frac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i). … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►uniformly with respect to bounded positive values of $x$ in each case. ►For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). …

33: Bibliography M

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

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Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.

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ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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A connection formula for the $q$-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt

Referenced by:

§17.6(ii), §17.6(ii).

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34: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. …

35: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►

§18.30(i) Associated Jacobi Polynomials

… ►For the confluent hypergeometric function $U$ see §13.2(i). … ►For Gauss’ hypergeometric function $F$ see (15.2.1). … ►See Ismail (2009, p. 46 ), where the $k$th corecursive polynomial is also related to an appropriate continued fraction, given here as its $n$th convergent, …

36: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►The approximations are expressed in terms of Whittaker functions $W_{\kappa ,\mu }\left(z\right)$ and $M_{\kappa ,\mu }\left(z\right)$ with $\mu =\frac{1}{4}$; compare §2.8(vi). … ►For related results see Langer (1934) and Sharples (1967, 1971). …

37: 33.23 Methods of Computation

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§33.23(i) Methods for the Confluent Hypergeometric Functions

►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►

§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). … ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

38: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)). … ►The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985). … ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. … ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. …

39: 13.2 Definitions and Basic Properties

… ► $M(a,b,z)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. … ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►Unless specified otherwise, however, $U(a,b,z)$ is assumed to have its principal value. … ►For $U(a,b,z)$ see (13.2.6). … ►

Kummer’s Transformations

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40: Bibliography

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Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:

ISSN 1047-5974, FullText

Referenced by:

§13.32(iii).

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Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:

ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt

Referenced by:

(25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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