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Using the extended Mittag-Leffler function, we introduce new extension to the extended Pochhammer symbol and gamma function. Applying this extended Pochhammer symbol, we also give an extension of the generalized hypergeometric function as well as some of its specific instances. We also define extension of the τ-Gauss hypergeometric function. Also included are integral and derivative formulas for this extended τ-Gauss hypergeometric function using the Mellin transform and fractional calculus methods. A few other intriguing and well-known findings are also provided by the new extended 'τ-Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

The resolution of isolated cases of the well-known quadratic transformation due to Kummer was thoroughly examined in two recent publications by Atia as well as Atia and Al-Mohaimeed. The objective of this paper is twofold. We establish generalizations of the quadratic transformation due to Kummer in the most general case in the first section, and in the second section, an effort is made to discuss the resolution of an isolated case of a quadratic transformation contiguous to that of Kummer established by Choi and Rathie. [ABSTRACT FROM AUTHOR]

We introduce the extended Pochhammer symbol and gamma function by additional extensions using the rRy function [6-]. We also provide extension of generalized hypergeometric function using this extended Pochhammer symbol and derive some of its particular cases. We present exten- sion of the T-Gauss hypergeometric function and also gives us a few more interesting and well-known results. Integral and derivative formulae for this extended r-Gauss hypergeometric function are also provided using fractional calculus techniques. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the Stirling numbers of the first kind and the second kind. By arranging these numbers into matrices, we generalize the Stirling matrices of the first kind and the second kind investigated by Cheon and Kim [Stirling matrix via Pascal matrix, Linear Algebra Appl. 329 (2001) 49-59]. Furthermore, we introduce generalizations of the Pascal matrix and the symmetric Pascal matrix with two real arguments, and generalize earlier results related to the Pascal matrices, Stirling matrices and matrices involving Bell numbers. [ABSTRACT FROM AUTHOR]

The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By the application of our method to known transformation formulas, we derive about forty hypergeometric identities, the majority of which are believed to be new. [ABSTRACT FROM AUTHOR]

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type. [ABSTRACT FROM AUTHOR]

The Topp-Leone family of univariate distributions has drawn considerable attention in recent years. In this article, we define a bivariate generalization of the Topp-Leone distribution. Moreover, we derive several results such as marginal and conditional distributions, joint moments, correlation coefficient, and Fisher information matrix. Furthermore, we derive the exact distributions of X + Y, X/(X + Y) and XY when X and Y follow bivariate Topp-Leone distribution. [ABSTRACT FROM AUTHOR]

The aim of this note is to provide a natural extension and generalisation of the well-known Gelfond constant eπ using a hypergeometric function approach. An extension is also found for the square root of this constant. Several known mathematical constants are also deduced in hypergeometric form from our newly introduced constant. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a result of Cuttler, Greene, Skandera, and Sra that characterizes the majorization order on Young diagrams in terms of nonnegative specializations of Schur polynomials. More precisely, we introduce a generalized notion of majorization associated to an arbitrary crystallographic root system |$\Phi $| and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system |$\Phi.$| We also conjecture a further generalization of this theorem in terms of Heckman–Opdam hypergeometric functions. [ABSTRACT FROM AUTHOR]

Exton [Ganita 54 (2003), 13-15] obtained numerous new quadratic transformations involving hypergeometric functions of order two and of higher order by applying various known classical summation theorems to a general transformation formula based on the Bailey transformation. We obtain a generalization of one of the Exton quadratic transformations. The results are derived with the help of a generalization of Dixon's summation theorem for the series 3F2 obtained earlier by Lavoie et al. Several interesting known as well as new special cases and limiting cases are also given. [ABSTRACT FROM AUTHOR]

Reduction formulas for the sum of products of hypergeometric functions can be traced back to Euler. This topic has intimate relations with summation and transformation formulas, contiguous relations, and algebraic properties of the (generalized) hypergeometric differential equation. Over the past several years important discoveries have been made in this topic by Gorelov, Ebisu, Beukers and Jouhet, and Feng, Kuznetsov and Yang. In this paper we give a generalization of the Feng, Kuznetsov, and Yang identity, also covering Ebisu's and Gorelov's formulas as particular cases. [ABSTRACT FROM AUTHOR]

In my joint papers with Iwaki and Koike, we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper, we discuss its generalization to the case of the hypergeometric differential equation of type (1 , 4). [ABSTRACT FROM AUTHOR]

We give a generalization of |$p$| -adic congruences for truncated period functions that were originally discovered for a class of hypergeometric functions by Bernard Dwork. [ABSTRACT FROM AUTHOR]

We introduce the |$q$| -Hahn PushTASEP—an integrable stochastic interacting particle system that is a three-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (totally asymmetric simple exclusion process). The transition probabilities in the |$q$| -Hahn PushTASEP are expressed through the |$_4\phi _3$| basic hypergeometric function. Under suitable limits, the |$q$| -Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the |$q$| -Hahn TASEP introduced by Povolotsky [ 37 ]. We establish Markov duality relations and contour integral formulas for the |$q$| -Hahn PushTASEP. In a |$q\to 1$| limit of our process we arrive at a random recursion, which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e. the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner. [ABSTRACT FROM AUTHOR]

We propose a new bilinear Hirota equation for |$\tau$| -functions associated with the |$E_8$| root lattice that provides a "lens" generalisation of the |$\tau$| -functions for the elliptic discrete Painlevé equation. Our equations are characterised by a positive integer |$r$| in addition to the usual elliptic parameters and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the |$E_8$| root lattice. We construct explicit |$W(E_7)$| -invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an analytical solution for an unsolved definite integral RC(m; n) from a 1915 paper of Srinivasa Ramanujan. We obtain our solution using the hypergeometric approach and an infinite series decomposition identity. Also, we give some generalizations of Ramanujan's integral RC(m; n) defined in terms of the ordinary hypergeometric function 2F3 with suitable convergence conditions. Moreover as applications of our result we obtain nine new infinite summation formulas associated with the hypergeometric functions 0F1, 1F2 and 2F3. [ABSTRACT FROM AUTHOR]

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We claim that, when parameters of the integrand are non-resonant, the D-module corresponding to Euler-Laplace integral is naturally isomorphic to GKZ hypergeometric system MA(δ) where A is a generalization of Cayley configuration. As a topological counterpart of this isomorphism, we establish an isomorphism between certain rapid decay homology group and holomorphic solutions of MA(δ). [ABSTRACT FROM AUTHOR]

We present three q-Taylor formulas with q-integral remainder. The two last proofs require a slight rearrangement by a well-known formula. The first formula has been given in different form by Annaby and Mansour. We give concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erdélyi, and for two of Srivastavas triple hypergeometric functions and other functions. All proofs are made in a similar style by using q-integration. We find some new formulas for fractional q-integrals including a series expansion. In the same way, the operator formulas by Srivastava and Manocha find a natural generalization. [ABSTRACT FROM AUTHOR]

By building a second-order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second-order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a 'generalized' Rodrigues formula is also established. [ABSTRACT FROM AUTHOR]

The aim in this note is to provide a generalization of an interesting entry in Ramanujan's notebooks that relate sums involving the derivatives of a function φ(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating 3F2 hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating 3F2(2) with applications, Bulletin of the Korean Mathematical Society 2012; 49: 621-633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences. [ABSTRACT FROM AUTHOR]

In this paper, we determine solutions for the Lévy-Leblond operator or a parabolic Dirac operator in terms of hypergeometric functions and spherical harmonics. We subsequently generalize our approach to a wider class of Dirac operators depending on 4 parameters. [ABSTRACT FROM AUTHOR]

Very recently Masjed-Jamei and Koepf [Axioms 2018, 7 (2), 38] established interesting and useful generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. The main aim of this paper is to establish eleven Eulerian's type integrals involving generalized hypergeometric functions by employing these theorems. Several special cases (known and unknown) have also been given. [ABSTRACT FROM AUTHOR]

We introduce here an entire function of order zero, representing a rapidly convergent power series which extends the generalized hypergeometric function pFq[z]. We introduce the infinite order hyper-Bessel operators and thereby obtain its differential equation and the eigen function property. Some contiguous function relations are also derived. We finally show that the Ramanujan's theorem and Kummer's first formula admit extension by means of this theory. [ABSTRACT FROM AUTHOR]

In this paper, we present further generalizations of gamma and beta functions by adding extra parameters to their integral representations. Then we introduce new generalizations of hypergeometric and confluent hypergeometric functions by using new beta function. We also obtain many interesting properties such as integral transforms, differentiation formulas, Mellin transforms, transformation formulas, differential and difference relations and a summation formula. [ABSTRACT FROM AUTHOR]

Explicit expressions for the hypergeometric series 2F1(-n,a; 2a±j; 2) and 2F1(-n; a;-2n±j;2) for positive integer n and arbitrary integer j are obtained with the help of generalizations of Kummer's second and third summation theorems obtained earlier by Rakha and Rathie. Results for |j|≤5 derived previously using different methods are also obtained as special cases. Two applications are considered, where the first summation formula is applied to a terminating 3F2(2) series and the confluent hypergeometric function ¹F1(x). [ABSTRACT FROM AUTHOR]

We study the mini-superspace semiclassical limit of the boundary three-point function in the Liouville field theory. We compute also matrix elements for the Morse potential quantum mechanics. An exact agreement between the former and the latter is found. We show that both of them are given by the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

We deal with the relation between the Mittag–Leffler functions and the sum of the hypervolume of generalizedn-balls. We derive from this result some simple properties of the Mittag–Leffler functions. We also define some new Abelian groups from the sum and the multiplication as well as the Pontryagin transform associated to them. We relate this approach to the calculus on measure chains. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of Whittaker integral transforms involving confluent hypergeometric functions as kernels. We present certain convolution products and derive the prerequisite axioms for generating the relevant spaces of Boehmians. We further give the definition and discuss the inverse problem of the Whittaker integral in a generalized sense. Moreover, the integral in question has been shown to be linear and consistent with the classical one. Certain results are also discussed in some details. [ABSTRACT FROM AUTHOR]

In this present paper, we deal with the more generalized functions involving Gauss hypergeometric k-functions Fk((a; k); (b; k); (c; k); z)... where | arg| < p, Re(u) > 0. This function reduces to a generalized gamma k-function recently defined by "[Mubeen et al.; Jour. Ineq. Spec. Funct. Volume 7 Issue 1(2016), pp.53-60 ]" when d = k. Also, it reduces to gamma k-function k when p = 1, d = k and b = c. In this paper, we obtain several properties for the newly defined generalized function Dk, its recurrence relations and asymptotic expansion for Dk where k > 0. We establish further generalization of Kobayashi's gamma k-function i.e., m,k(u; v), which is an improved version of Kobayashi's gamma function. [ABSTRACT FROM AUTHOR]

In this paper, we define further generalized hypergeometric k- functions, using a special case of Wright hypergeometric function. Some of the differential properties, integral representation, contiguous relations and differential formulas of the generalized hypergeometric k-functions 2R1;k(a; b; c; t; z) (where k > 0) are established. [ABSTRACT FROM AUTHOR]

Young et al., (2010) showed that due to gene length bias the popular Fisher Exact Test should not be used to study the association between a group of differentially expressed (DE) genes and a specific Gene Ontology (GO) category. Instead they suggest a test where one conditions on the genes in the GO category and draws the pseudo DE expressed genes according to a length-dependent distribution. The same model was presented in a different context by Kazemian et al., (2011) who went on to offer a dynamic programming (DP) algorithm to exactly compute the significance of the proposed test. Here we point out that while valid, the test proposed by these authors is no longer symmetric as Fisher's Exact Test is: one gets different answers if one conditions on the observed GO category than on the DE set. As an alternative we offer a symmetric generalization of Fisher's Exact Test and provide efficient algorithms to evaluate its significance. [ABSTRACT FROM AUTHOR]

Within the Askey scheme of hypergeometric orthogonal polynomials, Racah polynomials stay on the top of the hierarchy and they generalize all of the discrete hypergeometric orthogonal polynomials. In this paper, we investigate asymptotic behaviors of Racah polynomials with varying parameters when the polynomial degree tends to infinity. Using the difference equation technique developed in our earlier papers, we obtain an asymptotic formula in the outer region via ratio asymptotics and then derive asymptotic formulas in the oscillatory region via a matching method. Our asymptotic formulas are explicitly given in terms of the polynomial degree, variable and parameters, using elementary functions such as logarithmic, exponential and rational functions. By taking limits, our results also yield asymptotic formulas for orthogonal polynomials in the lower hierarchy of the Askey scheme such as Hahn polynomials and Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors. [ABSTRACT FROM AUTHOR]

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of Q contains a quaternion algebra. [ABSTRACT FROM AUTHOR]

In this paper, we present several generalizations and refinements for the asymptotic formulas of Gaussian and generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper we discuss exponential functionals of the multi-factor variance gamma (VG) process. This process is a generalization of the well-known VG process. We obtain analytically the expectations of the considered functionals. Derived formulas are based on values of generalized hypergeometric functions. Applications of the established results to option pricing are given. [ABSTRACT FROM AUTHOR]

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 typeand 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. [ABSTRACT FROM AUTHOR]

We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner–Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials. [ABSTRACT FROM AUTHOR]

An integral representation for the generalized hypergeometric function unifying known representations via generalized Stieltjes, Laplace, and cosine Fourier transforms is found. Using positivity conditions for the weight in this representation, various new facts regarding generalized hypergeometric functions, including complete monotonicity, log-convexity in upper parameters, monotonicity of ratios, and new proofs of Luke's bounds are established. In addition, two-sided inequalities for the Bessel type hypergeometric functions are derived with the use of their series representations. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

This paper aims to provide a natural generalization and unification of a series of multiple integral representations for special classes of hypergeometric polynomials recently obtained by several authors. This generalization is obtained by considering a very large family of hypergeometric polynomials. The multiple integral representations given in this paper may be viewed as linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector version of the Nova q-addition. These convergence regions are given in a purely formal way, extending the results of Karlsson (1976). The Γq-function and the q-binomial coefficients, which are used in the proofs, are adjusted accordingly. Furthermore, limits and special cases for the new functions, e.g., q-Lauricella functions and q-Horn functions, are pointed out. [ABSTRACT FROM AUTHOR]

We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the first kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed. [ABSTRACT FROM PUBLISHER]

New explicit as well as manifestly symmetric three-term summation formulas are derived for the Clausenian hypergeometric series 3 F 2 ( 1 ) with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P.W. Karlsson. [ABSTRACT FROM AUTHOR]

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers. [ABSTRACT FROM AUTHOR]

We obtain the asymptotic behavior of the zeros of a class of generalized hypergeometric polynomials. For this purpose, we make use of a Mehler–Heine type formula for these polynomials. We illustrate these results with numerical experiments and some figures. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish a quite general transformation involving the generalized hypergeometric function. Extensions of Kummer's first transformation, Gauss and Kummer summation, and its contiguous results are then applied to obtain a new class of summation formulas involving the generalized hypergeometric 2F2 polynomial, which have not previously appeared in the literature. A few well-known results obtained earlier by Kim et al. (Int. J. Math. Math. Sci. 2010:309503, 2010; Integral Transforms Spec. Funct. 23(6):435-444, 2012) and Exton have been obtained as special cases of our main findings. [ABSTRACT FROM AUTHOR]

We define the generalized hypergeometric polynomial of degree N as follows: P N ( α 1 , . . . , α p ; β 1 , . . . , β q ; z ) = ∑ m = 0 N [ ( − N ) m ( α 1 ) m ⋅ ⋅ ⋅ ( α p ) m z N − m m ! ( β 1 ) m ⋅ ⋅ ⋅ ( β q ) m ] = z N F q p + 1 ( − N , α 1 , . . . , α p ; β 1 , . . . , β q ; 1 / z ) . Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p + q parameters α j and β k are arbitrary (“generic”, possibly complex) numbers, ( α ) m is the Pochhammer symbol and F q p + 1 ( α 0 , α 1 , . . . , α p ; β 1 , . . . , β q ; z ) is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζ n of this polynomial. We moreover manufacture an N × N matrix L ̲ in terms of the 1 + p + q parameters N , α j , β k characterizing this polynomial, and of its N zeros ζ n , and we show that it features the N eigenvalues λ m = m ∏ k = 1 q ( − β k + 1 − m ) , m = 1 , . . . , N . These N eigenvalues depend only on the q parameters β k , implying that the N × N matrix L ̲ is isospectral for variations of the p parameters α j ; and they clearly are integer (or rational ) numbers if the q parameters β k are themselves integer (or rational ) numbers: a nontrivial Diophantine property. [ABSTRACT FROM AUTHOR]

Recently, Srivastava et al. (2012) [11] introduced and initiated the study of many interesting fundamental properties and characteristics of a certain pair of potentially useful families of the so-called generalized incomplete hypergeometric functions. Ever since then there have appeared many closely-related works dealing essentially with notable developments involving various classes of generalized hypergeometric functions and generalized hypergeometric polynomials, which are defined by means of the corresponding incomplete and other novel extensions of the familiar Pochhammer symbol. Here, in this sequel to some of these earlier works, we first derive two remarkably distinct classes of hypergeometric generating functions and then apply each of them in deducing a family of linear, bilinear and bilateral (or mixed multilateral) generating functions. We also consider various (known or new) special cases and consequences of the results presented in this paper. [ABSTRACT FROM AUTHOR]

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

Abstract: The Fourier Transform of a quartic Gaussian, , is important in the theory of radial basis functions with . We show that the transform is the exact sum of two generalized hypergeometric functions . This implies that is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree. Using the method of steepest descent for integrals, we derive the first five terms in an asymptotic series in powers of . Through a simple hyperasymptotic analysis, we show that the magnitude of the error of the optimally-truncated series [“superasymptotic” error] is . The lowest order approximation, shows that the transform is an exponentially-decaying oscillation. We show that the steepest descent method [2,27] (Bender and Orszag, 1978; Miller, 2006) [26,33] (Lauwerier, 1966; Ursell, 1965) can be applied to the transform of for general integer n and give the first two terms. Another theme is that modern computer technology has simplified the steepest descent method. The steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function where is the logarithm of the integrand, and z is the large parameter (here proportional to ). The task of inverting to obtain a series in powers of w for the metric factor after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete Maple code. [Copyright &y& Elsevier]