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Page 3706 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
The generating function is also used to express a number of integrals involving the LFG polynomials as finite sums of parabolic cylinder functions.” 91g:33012 33C45 41A10 41A60 Sharapudinov, I.  ...  For details about the above hypergeometric 33 SPECIAL FUNCTIONS 3706 family of polynomials a good reference would be to a book by T. S.  ... 

A numerical strategy for efficient modeling of the equatorial wave guide

A. J. Majda, B. Khouider
2001 Proceedings of the National Academy of Sciences of the United States of America  
Parabolic Cylinder Functions, Hermite Polynomials, and Gauss-Hermite Quadrature The parabolic cylinder functions are given by where H N () are the Hermite polynomials.  ...  The plan for the rest of this paper is the following: first, basic properties of parabolic cylinder functions, Hermite polynomials, and Gauss-Hermite quadrature are reviewed, which are necessary for the  ...  This research of A.M. is partially supported by grants from the National Science Foundation and the Office of Naval Research. B.K. is supported as a postdoctoral fellow through these grants.  ... 
doi:10.1073/pnas.98.4.1341 pmid:11171952 pmcid:PMC29258 fatcat:5dlq2tcsize7hlho4tli4wqlie

A numerical strategy for efficient modeling of the equatorial wave guide

A. J. Majda
2001 Proceedings of the National Academy of Sciences of the United States of America  
Parabolic Cylinder Functions, Hermite Polynomials, and Gauss-Hermite Quadrature The parabolic cylinder functions are given by where H N () are the Hermite polynomials.  ...  The plan for the rest of this paper is the following: first, basic properties of parabolic cylinder functions, Hermite polynomials, and Gauss-Hermite quadrature are reviewed, which are necessary for the  ...  This research of A.M. is partially supported by grants from the National Science Foundation and the Office of Naval Research. B.K. is supported as a postdoctoral fellow through these grants.  ... 
doi:10.1073/pnas.041583698 pmid:11171952 pmcid:PMC29258 fatcat:h7zv53f6p5cp5dj6ugb7mc6uha

Smooth surface interpolation using patches with rational offsets [article]

Miroslav Lávička, Zbyněk Šír, Jan Vršek
2016 arXiv   pre-print
We present a new method for the interpolation of given data points and associated normals with surface parametric patches with rational normal fields.  ...  Our construction is based on the isotropic model of the dual space to which the original data are pushed.  ...  We thank to all referees for their valuable comments, which helped us to improve the paper.  ... 
arXiv:1602.01224v2 fatcat:27sqblroifbmhn3svpdwhhhkx4

Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations [article]

Peter A. Clarkson and Andrew P. Bassom (Department of Mathematics, University of Exeter, Exeter, U.K.)
1994 arXiv   pre-print
Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder functions as well as rational solutions will be derived  ...  A suitable factorisation of (2) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable z_n and the limits of these solutions yield rational  ...  In addition he is grateful to the staff and students of New College, UNSW for their provision of a Visiting Fellowship and to the School of Mathematics for their hospitality.  ... 
arXiv:solv-int/9412002v1 fatcat:r2lpt5xmy5fbxb2tbokudp2xg4

Page 1285 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
, we derive a dependence on Hermite polynomials and error functions.”  ...  L. 87c:34004 Functions on a parabolic cylinder with a negative integer index. (Russian) Differentsial'nye Uravneniya 21 (1985), no. 11, 2001-2003, 2024.  ... 

On bounds for solutions of monotonic first order difference-differential systems

Javier Segura
2012 Journal of Inequalities and Applications  
Many special functions are amenable to this analysis, and we give several examples of application: modified Bessel functions, parabolic cylinder functions, Legendre functions of imaginary variable and  ...  are sharp both as a function of n and x.  ...  functions (see Section 4.1.2 for the case of parabolic cylinder functions).  ... 
doi:10.1186/1029-242x-2012-65 fatcat:3gcup6lfibhordqyqcctufkfsa

Products & Materials

1987 Science  
Nuclear Associates. Circle 666. program. ESS operates on VAX and UNIX computers and workstations. New MethodsResearch. Circle 533.  ...  , Euler, Debye, and Spence), exponential integral and related functions (including cosine and sine integrals and hyperbolic extensions, Fresnel integrals, and parabolic cylinder functions), periodic functions  ... 
doi:10.1126/science.238.4833.1597 pmid:17784302 fatcat:tevonn5vuvhxlplp5d4xxqgkni

Formal matched asymptotics for degenerate Ricci flow neckpinches

Sigurd B Angenent, James Isenberg, Dan Knopf
2011 Nonlinearity  
In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.  ...  Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on S^m, for all m≥ 3.  ...  The associated eigenfunctions are the Hermite polynomials h k (σ).  ... 
doi:10.1088/0951-7715/24/8/007 fatcat:b32wlcbowveuhbwn3tf3nnfexe

Operational vs. Umbral Methods and Borel Transform [article]

Giuseppe Dattoli, Silvia Licciardi
2019 arXiv   pre-print
We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.  ...  Differintegral methods, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only.  ...  Acknowledgements The work of Dr. S. Licciardi was supported by an Enea Research Center individual fellowship. Author Contributions  ... 
arXiv:1908.04160v1 fatcat:7szretfsf5djpddjuz34ox7eya

New exact solutions of the discrete fourth Painleve´equation

Andrew P. Bassom, Peter A. Clarkson
1994 Physics Letters A  
A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable z_n.  ...  It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can  ...  In addition he is grateful to the staff and students of New College, UNSW for their provision of a Visiting Fellowship and to the School of Mathematics for their hospitality.  ... 
doi:10.1016/0375-9601(94)91294-7 fatcat:vuodrsmfvjfjzptuqyhd76ppv4

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

Galina V. Filipuk, Peter A. Clarkson
2008 Studies in applied mathematics (Cambridge)  
will become new members of the community of special functions".  ...  Further representations are given of the associated special polynomials in terms of Schur functions.  ...  A family of τ -functions In this section we give a description of some of the rational solutions of the symmetric A (1) 2n system (1.1) in terms of the Schur functions.  ... 
doi:10.1111/j.1467-9590.2008.00410.x fatcat:7fqbieo7cvckzn4zbe5fwc4d3q

Nonlinear coherent states associated with a measure on the positive real half line [article]

S. Twareque Ali, Zouhaïr Mouayn, Khalid Ahbli
2018 arXiv   pre-print
We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated coherent states transform is discussed.  ...  We illustrate this construction with the example of the measure r^β e^-rdr, which leads to a new generalization of the true-polyanalytic Bargmann transform.  ...  + 1) −1 |D −β (ix √ 2)| −2 dx where D a (•) is the parabolic cylinder function.  ... 
arXiv:1812.11620v1 fatcat:kmefovj4f5awtecpclsp3oejo4

A program for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations

Dimitris M Christodoulou, James Graham-Eagle, Qutaibeh D Katatbeh
2016 Advances in Difference Equations  
The validity of the method depends on one's ability to transform a given differential equation to its simplest possible form, so a program must be executed that involves transformations of both variables  ...  We derive a new criterion for deducing the intervals of oscillatory behavior in the solutions of ordinary second-order linear homogeneous differential equations from their coefficients.  ...  First form of the parabolic cylinder equation We call the equation where n is a constant [], the first form of the parabolic cylinder differential equation.  ... 
doi:10.1186/s13662-016-0774-x fatcat:vkd4ibglnnfv3do3hietmejnha

Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations

Peter A. Clarkson
2006 Computational methods in Function Theory  
In this paper the structure of the roots of these special polynomials, and the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations, is studied and it is shown  ...  Rational solutions of the second, third and fourth Painlevé equations (P II -P IV ) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order  ...  expressible in terms of parabolic cylinder functions D ν (ξ), or a special case of the Whittaker functions M κ,µ (ζ) and W κ,µ (ζ) (cf  ... 
doi:10.1007/bf03321618 fatcat:6kp4ohmfi5haho2pmnl44ny2ne
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