relation to parabolic cylinder functions

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11: 16.18 Special Cases

§16.18 Special Cases

… ►This is a consequence of the following relations: …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

12: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

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Ultraspherical

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Laguerre

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Hermite

… ►For the parabolic cylinder function $U(a,z)$, see §12.2. …

13: Bibliography S

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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.

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ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

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3rd item.

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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On parabolic cylinder functions. J. Inst. Math. Appl. 4 (1), pp. 106–112.

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Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§12.16.

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14: Bibliography B

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A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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1st item.

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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2nd item, 1st item.

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The asymptotic analysis of canonical problems in high-frequency scattering theory. II. The circular and parabolic cylinders. Proc. Cambridge Philos. Soc. 74, pp. 313–332.

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Document, MathReview (Y. M. Chen)

Referenced by:

§12.17.

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Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions. Proc. Amer. Math. Soc. 114 (4), pp. 1025–1032.

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ISSN 0002-9939, FullText, MathReview, ZentralBlatt

Referenced by:

§12.16.

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15: 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). … ►Then as $h\to +\mathrm{\infty }$ … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►For related results see Langer (1934) and Sharples (1967, 1971). …

16: 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 parabolic cylinder function $U$ see §12.2(i). … ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. … ►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, …

17: 12.8 Recurrence Relations and Derivatives

§12.8 Recurrence Relations and Derivatives

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§12.8(i) Recurrence Relations

… ►(12.8.1)–(12.8.4) are also satisfied by $\overline{U}(a,z)$. … ►

§12.8(ii) Derivatives

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12.8.10 $$\frac{d^m}{{dz}^m}\left({\mathrm{e}}^{-\frac{1}{4}{z}^2}U(a,z)\right)={(-1)}^m{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(a-m,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§12.8(ii)

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http://dlmf.nist.gov/12.8.E10

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See also:

Annotations for §12.8(ii), §12.8 and Ch.12

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18: Bibliography L

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Functions on a parabolic cylinder with a negative integer index. Differ. Uravn. 21 (11), pp. 2001–2003, 2024 (Russian).

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Notes:

In Russian.

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ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§12.13(i).

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Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§16.4(iii).

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Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§16.4(iii).

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

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19: Bibliography M

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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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Tables of Weber Parabolic Cylinder Functions. Her Majesty’s Stationery Office, London.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.16, §10.39, §12.14(i), §12.14(ii), §12.14(iv), §12.14(v), §12.14(v), §12.14(vi), §12.14(vii), §12.14(viii), §12.14(x), §12.14(x), 2nd item, §12.2(i), §12.2(ii), §12.2(iii), §12.2(iv), §12.2(v), §12.2(vi), §12.2(vi), §12.4, §12.5(i), §12.5(ii), §12.5(iii), §12.7(i), §12.7(ii), §12.7(iii), §12.7(iii), §12.7(iv), §12.8(i), Ch.12.

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Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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Document, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§12.13(ii), §12.17.

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Elastodynamics in parabolic cylinders. Z. Angew. Math. Phys. 39 (5), pp. 748–752.

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ISSN 0044-2275, Document, MathReview (Sergej M. Zverev), ZentralBlatt

Referenced by:

§12.17.

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Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.

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MathReview, ZentralBlatt

Referenced by:

7th item.

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20: 32.11 Asymptotic Approximations for Real Variables

… ►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{U}^2(-\nu -\frac{1}{2},\sqrt{2}x)$ as $x\to +\mathrm{\infty }$, where $h$ $(\ne 0)$ is a constant. Conversely, for any $h$ $(\ne 0)$ there is a unique solution $w_h(x)$ of (32.11.29) that is asymptotic to $h{U}^2(-\nu -\frac{1}{2},\sqrt{2}x)$ as $x\to +\mathrm{\infty }$. Here $U$ denotes the parabolic cylinder function (§12.2). …