Hypergeometric Functions Of Unit Argument Research Articles

New orthogonality relationships of the Gaunt coefficients

A new analytical formula for the Gaunt coefficients is derived using expressions of Clebsch Gordan coefficients in terms of the generalized hypergeometric functions of unit argument and binomial coefficients. In addition, new sum expressions and orthogonality relations containing the Gaunt coefficients are written. These expressions are used to test the accuracy of the numerical calculation for the Gaunt coefficients. These formulae are coded in the Mathematica programming language, and the Gaunt coefficients are calculated numerically and compared with the values from the literature.

New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

New product and linearization formulae of Jacobi polynomials of certain parameters

In this research article, a new product formula of Jacobi polynomials of certain parameters is established. This formula is expressed in terms of a terminating hypergeometric function of the type and it generalizes a formula which connects explicitly the squares of two ultraspherical polynomials with different parameters. Thanks to symbolic algebraic computation, and in particular, the celebrated algorithms of Zeilberger and Petkovsek-van Hoeij, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence some new product and linearization formulae of Jacobi polynomials of certain parameters are deduced. The latter formulae are used to obtain new formulae for some definite integrals.

On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kinds

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

Linearization formulae for certain Jacobi polynomials

In this article, some new linearization formulae of products of Jacobi polynomials for certain parameters are derived. These new derived formulae are expressed in terms of hypergeometric functions of unit argument, and they generalize some existing formulae in the literature. With the aid of some standard formulae and also by employing symbolic algebraic computation, and in particular Zeilberger’s algorithm, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence several linearization formulae of products of Jacobi polynomials for special parameters free of hypergeometric functions are deduced.

New formulae for the high-order derivatives of some Jacobi polynomials: an application to some high-order boundary value problems.

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Master integrals for fermionic contributions to massless three-loop form factors

In this Letter we continue the calculation of master integrals for massless three-loop form factors by giving analytical results for those integrals which are relevant for the fermionic contributions proportional to NF2, NF⋅N, and NF/N. Working in dimensional regularisation, we express one of the integrals in a closed form which is exact to all orders in ϵ, containing Γ-functions and hypergeometric functions of unit argument. In all other cases we derive multiple Mellin–Barnes representations from which the coefficients of the Laurent expansion in ϵ are extracted in an analytical form. To obtain the finite part of the three-loop quark and gluon form factors, all coefficients through transcendentality six in the Riemann ζ-function have to be included.

Two-loop quark and gluon form factors in dimensional regularisation

We compute the two-loop corrections to the massless quark form factor γ∗→qq¯ and gluon form factor H→gg to all orders in the dimensional regularisation parameter ε=(4−d)/2. The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as Γ-functions and generalised hypergeometric functions of unit argument. Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in ε. We provide expansions of the form factors to order ε2, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.

Some connection and linearization problems for polynomials in and beyond the Askey scheme

The connection problem is considered in a hypergeometric function framework for (i) the two most general families of polynomials belonging to the Askey scheme (Wilson and Racah), and (ii) some generalized Laguerre and Jacobi polynomials falling outside that scheme (Sister Celine, Cohen and Prabhakar–Jain), which are relevant to the study of quantum-mechanical systems and include as particular cases, the generalizations of the classical families with Sobolev-type orthogonality. In addition, using the same method three new linearization-like formulae for the Gegenbauer polynomials are also derived: a linearization formula that generalizes the m= n case of Dougall's formula, the analogue of the m= n case of Nielsen's inverse linearization formula for Hermite polynomials, and a connection formula for the squares. Closed analytical formulae for the corresponding connection and linearization coefficients are given in terms of hypergeometric functions of unit argument, which at times can be further simplified and expressed as single hypergeometric terms.

A proof of the equivalence of two formulas for the matrix elements of the Morse potential

Two important formulas for the matrix elements of the Morse potential are shown to be equivalent. The proof rests on certain theorems relating generalized hypergeometric functions of unit argument.

Associated Wilson polynomials

From a contiguous relation obtained by Wilson for terminating 2-balanced very well-poised9 F 8 hypergeometric functions of unit argument, we derive a pair of three term recurrence relations for very well-poised7 F 6's. From these we obtain solutions to the recurrence relation for associated Wilson polynomials and spectral properties of the corresponding Jacobi matrix. A calculation of the basic weight function yields a generalization of Dougall's theorem.

Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential Equations

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. 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Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Recurrence Relations for Hypergeometric Functions of Unit Argument

We show that the generalized hypergeometric function \[ {P_n}:{ = _{p + 3}}{F_{p + 2}}\left ( {\left . {\begin {array}{*{20}{c}} { - n,n + \lambda ,{a_p},1} \\ {{b_{p + 2}}} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] satisfies a nonhomogeneous recurrence relation of order $p + \sigma$, where $\sigma = 0$ when $_{p + 3}{F_{p + 2}}(1)$ is balanced, and $\sigma = 1$ otherwise. Also, for \[ {U_n}: = \frac {{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambda )}_n}}}{ _{q + 2}}{F_{q + 1}}\left ( {\left . {\begin {array}{*{20}{c}} {n + {c_{q + 2}}} \\ {n + {d_q},2n + \lambda + 1} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] a homogeneous recurrence relation of order $q + 1$ is given.

Recurrence relations for hypergeometric functions of unit argument

We show that the generalized hypergeometric function \[ P n : = p + 3 F p + 2 ( − n , n + λ , a p , 1 b p + 2 | 1 ) ( n ⩾ 0 ) {P_n}:{ = _{p + 3}}{F_{p + 2}}\left ( {\left . {\begin {array}{*{20}{c}} { - n,n + \lambda ,{a_p},1} \\ {{b_{p + 2}}} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0) \] satisfies a nonhomogeneous recurrence relation of order p + σ p + \sigma , where σ = 0 \sigma = 0 when p + 3 F p + 2 ( 1 ) _{p + 3}{F_{p + 2}}(1) is balanced, and σ = 1 \sigma = 1 otherwise. Also, for \[ U n := ( c q + 1 ) n ( d q ) n ( n + λ ) n q + 2 F q + 1 ( n + c q + 2 n + d q , 2 n + λ + 1 | 1 ) ( n ⩾ 0 ) {U_n}: = \frac {{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambda )}_n}}}{\,_{q + 2}}{F_{q + 1}}\left ( {\left . {\begin {array}{*{20}{c}} {n + {c_{q + 2}}} \\ {n + {d_q},2n + \lambda + 1} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0) \] a homogeneous recurrence relation of order q + 1 q + 1 is given.

Special topics in the quantum theory of angular momentum

Two special topics in the quantum theory of angular momentum are discussed in this article. They are: (i) the relationship between the coupling and recoupling coefficients (for two and three angular momenta, respectively) and sets of generalized hyper-geometric functions of unit argument; and (ii) the ‘non-trivial’ or polynomial zeros of angular momentum coefficients and their classification.

Computation of angular momentum coefficients using sets of generalised hypergeometric functions

We demonstrate the necessity for sets of generalised hypergeometric functions of unit argument to account for the known symmetries of the 3- Wj and 6- Wj coefficients. These p F q (1) sets provide new algorithms which enable accurate and faster computation of the 3- Wj and 6- Wj coefficients, than what has been possible hitherto.