Laplace Transform of Some Hypergeometric Functions

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to , hypergeometric functions, and show that one just reduces to the previous cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

Many distributions in multivariate analysis can be expressed in a form involving hypergeometric functions $_pF_q$ of matrix argument e.g. the noncentral Wishart $(_0F_1)$ and the noncentral multivariate $F(_1F_1)$. For an exposition of distributions in this form see James [9]. The hypergeometric function $_pF_q$ has been defined by Constantine [1] as the power series representation \begin{equation*}\tag{1.1} _pF_q(a_1,\cdots, a_p; b_1,\cdots, b_q; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a_1)_\kappa\cdots(a_p)_\kappa}{(b_1)_\kappa\cdots (b_q)_\kappa} \frac{C_\kappa (R)}{k!}\end{equation*} where $a_1,\cdots, a_p, b_1,\cdots, b_q$ are real or complex constants, $(a)_\kappa = \mathbf{prod}^m_{i=1}(a - \frac{1}{2}(i - 1))_{k_i},\quad (a)_n = a(a + 1)\cdots (a + n - 1)$ and $C_\kappa(R)$ is the zonal polynomial of the $m \times m$ symmetric matrix $R$ corresponding to the partition $\kappa = (k_1, k_2,\cdots, k_m), k_1 \geqq k_2 \geqq \cdots \geqq k_m$, of the integer $k$ into not more than $m$ parts. The functions defined by (1.1) are identical with the hypergeometric functions defined by Herz [5] by means of Laplace and inverse Laplace transforms. For a detailed discussion of hypergeometric functions and zonal polynomials, the reader is referred to the papers [1] of Constantine and [7], [8], [9] of James. From a practical point of view, however, the series (1.1) may not be of great value. Although computer programs have been developed for calculating zonal polynomials up to quite high order, the series (1.1) may converge very slowly. It appears that some asymptotic expansions for such functions must be obtained. It is well known that asymptotic expansions for a function can in many cases be derived using a differential equation satisfied by the function (see e.g. Erdelyi [4]), and so, with this in mind, a study of differential equations satisfied by certain hypergeometric functions certainly seems justified. In this paper a conjecture due to A. G. Constantine is verified i.e. it is shown that the function \begin{equation*}\tag{1.2} _2F_1(a, b; c; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a)_\kappa(b)_\kappa}{(c)_\kappa} \frac{C_\kappa(R)}{K!}\end{equation*} satisfies the system of partial differential equations \begin{align*} \tag{1.3} R_i(1 &- R_i)\partial^2F/\partial R_i^2 + \{ c - \frac{1}{2}(m - 1) - (a + b + 1 - \frac{1}{2}(m - 1))R_i \\ &+\frac{1}{2} \sum^m_{j=1,j\neq i}\lbrack R_i(1 - R_i)/(R_i - R_j) \rbrack\}\partial F/\partial R_i \\ &-\frac{1}{2} \sum^m_{j=1,j\neq i} \lbrack R_j(1 - R_j)/(R_i - R_j) \rbrack\partial F/\partial R_j = abF \quad (i = 1,2, \cdots, m)\end{align*} where $R_1, R_2,\cdots, R_m$ are the latent roots of the complex symmetric $m \times m$ matrix $R$. When $m = 1$, the system (1.3) clearly reduces to the classical hypergeometric equation. It appears difficult to establish this conjecture directly, and the method used has necessitated a section devoted to a summary of the argument involved (Section 3). The main result in the paper is summarized in Theorem 3.1 of this section. Section 4 contains proofs referred to in Section 3. Using the fact that $C_\kappa (R)$ satisfies the partial differential equation (James [10])\begin{equation*}\tag{1.4} \sum^m_{i=1} R_i^2\partial^2y/\partial R_i^2 + \sum^m_{j=1,j\neq i}\lbrack R_i^2/(R_i - R_j) \rbrack\partial y/\partial R_i = \sum^m_{i=1}k_i(k_i + m - i - 1)y,\end{equation*} James and Constantine [11] have obtained the effects of certain differential operators on $C_\kappa(R)$. These results are given in Section 2 and are used in many proofs in Section 4. Section 5 is probably of most interest statistically, for here systems of partial differential equations similar to (1.3) are given for $_1F_1(a; c; R)$ and $_0F_1(c; R)$. These two functions occur often in multivariate distributions. The differential equations for $_1F_1(a; c; R)$ have been used by Constantine [3] to obtain an asymptotic expansion for the noncentral likelihood ratio criterion, and by the author [12] to obtain asymptotic distributions of Hotelling's generalized $T_0^2$ statistic, Pillai's $V^{(m)}$ criterion, and for the largest latent root of the covariance matrix. The system for $_0F_1(c; R)$ is a generalization of that given by James [6] for $_0F_1(m/2; R)$.

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

These lecture notes discuss some of the basics of elliptic hypergeometric functions. These are fairly recent generalizations of ordinary hypergeometric functions. In this chapter we first discuss both ordinary hypergeometric functions and elliptic functions, as you need to know both to define elliptic hypergeometric series. We subsequently discuss some of the important properties these series satisfy, in particular we consider the biorthogonal functions found by Spiridonov and Zhedanov, both with respect to discrete and continuous measure. In doing so we naturally encounter the most important evaluation and transformation formulas for elliptic hypergeometric series, and for the associated elliptic beta integral.

Special functions such as hypergeometric function and Laguerre polynomials are particular mathematical functions which are having a great significant importance in MIMO wireless systems. From the fast decade, MIMO systems are developed to increase the data rate and reliability in wireless communication networks. Capacity is one of the parameter to measure the data rate. In this paper, a closed form expression is derived analytically based on the confluent hypergeometric function for MIMO systems

In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z),g(z) in powers of z so that f(z) and \(f\left( z \right)\log z + g\left( z \right)\) satisfy a hypergeometric equation under a special choice of parameters, we prove that the series \(q\left( z \right) = ze^{{{g\left( {Cz} \right)} \mathord{\left/ {\vphantom {{g\left( {Cz} \right)} {f\left( {Cz} \right)}}} \right. \kern-\nulldelimiterspace} {f\left( {Cz} \right)}}}\) in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's p-adic technique.

In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed.

An integral transform $\bar f(s) = \int f(t)\varphi (s,t)dt$ can be averaged over the convex hull of $\{ s_1 , \cdots ,s_k \} $ to produce an analogous function $\bar F\{ s_1 , \cdots ,s_k \} $ of several real or complex variables. The question arises whether it is legitimate to take the average under the integral sign, so that $\bar F\{ s_1 , \cdots ,s_k \} = \int f(t)\Phi (s_1 , \cdots ,s_k ;t)dt$, where $\Phi $ is the corresponding average of $\varphi $. Conditions for the validity of this equation are of interest in the theory of special functions because the kernel $\Phi $ may be a Bessel function, elliptic integral, or other hypergeometric function when $\varphi $ is the kernel of a Laplace, Fourier, or Stieltjes transformation. Sufficient conditions of validity are established in the case of these three transformations and the inverse Laplace, Fourier, and Mellin transformations. The averages have some but not all of the operational properties of the corresponding ordinary transforms. Some examples are given involving various special functions.

PurposeTo study the multiplicative perturbation of local C‐regularized cosine functions associated with the following incomplete second order abstract differential equations in a Banach space X u″(t)=A(I+B)u(t), u(0)=x, u′(0)=y,(*) where A is a closed linear operator on X and B is a bounded linear operator on X.Design/methodology/approachThe multiplicative perturbation of exponentially bounded regularized C‐cosine functions is generally studied by the Laplace transformation. However, C‐cosine functions might not be exponentially bounded, so that the new method for the multiplicative perturbation of the nonexponentially bounded regularized C‐cosine functions should be applied. In this paper, the property of regularized C‐cosine functions is directly used to obtain the desired results.FindingsThe new results of the multiplicative perturbations of the nonexponentially bounded C‐cosine functions are obtained.Originality/valueThe new techniques differing from those given previously in the literature are employed to deduce the desired conclusions. The results can be applied to deal with incomplete second order abstract differential equations which stem from cybernetics, engineering, physics, etc.

We derive Lyapunov-type inequalities for third-order half-linear difference equations. The results are significant as the existing work on Lyapunov-type inequalities for difference equations is limited only to even order cases. As the techniques for even order difference equations are not applicable to odd order difference equations, the methods used here are novel. The resulting inequalities utilize and . New results for linear difference equations are obtained as a special case and are further extended to more general linear equations. Moreover, brief discussions are also given for third-order backward difference equations and dynamic equations on time scales.

Oscillation criteria for a forced mixed type Emden–Fowler equation with impulses

We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. Here these ideas are applied to the (continuous) Askey-Wilson polynomials, introduced by Askey and Wilson, to obtain a strikingly simple derivation of their orthogonality relations. 1. In t roduct ion . In [1] the authors and A.K. Agarwal introduced symmetry techniques for the study of families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. The fundamental objects in this study are the recurrence relations obeyed by the families: generating functions and identities for each family are characterized in terms of the recurrence relations. The functions studied in [1] are of the form ^ / a i , . . . , a r _ y > (ai;q)k'(ar;q)k z k V 6 i , . . . , 6 8 ' ' / f^Q(bi;q)k'(b8;q)k{q;q)k and many-variable extension. Here , . _ J 1 if k = 0 ( f l ; g , ' ( l a ) ( l g a ) . . . ( l c * 1 o ) if fc = 1,2,... and \q . ( , , . ) ( . ( I ^ ) ( I ^ E.G. KALNINS AND W. MILLER, JR. 225 Here Eff(z) = f(qz). Relation (2.1A) follows from T(az;q)J-;q) = J ^ ( l q)(aqz;q)k.1 (^— ;q) \Z /k q' Z / k-l and (2.IB) follows from KaZ;q)k(±q)k = (1 a b q ^ ^ q ) ^ ^ q ) ^ The first relation was pointed out by Askey and Wilson [2]; we have not found (2. IB) in the literature. Consider relation (2.IB); it suggests that there should be an operator Aaq-^,bq-^,cq,dq) . Aa,b,c,d) T , , a A l± mapping Cn t o <f)n . Indeed we find l(cq ^,dq^iaq' ^ibq^)ji{aq-^ ,bq-^,cqAq) (2.3) ^(l-Ht-^) ,(aM

1. Introduction to Differential Equations. 2. First-Order Equations. 3. Second and Higher-Order Linear Differential Equations. 4. Some Physical Applications of Linear Differential Equations. 5. Power Series Solutions of Differential Equations. 6. Laplace Transforms. 7. Introduction to Systems of Linear Differential Equations and Applications. 8. Numerical Methods. 9. Matrix Methods for Systems of Differential Equations. 10. Nonlinear Equations and Stability. 11. Fourier Series and Boundary Value Problems. 12. Partial Differential Equations. Appendices.

Apresenta-se a aplicação dos operadores de fatorização para a resolução das principais equações diferenciais li­neares de segunda ordem da Física Matemática como alternativa para o método de Frobenius. Em equações mais simples, a fatoração reduz a equação diferencial de segunda ordem a um par de equações diferenciais de primeira ordem, cada qual forne­cendo uma das soluções linearmente independentes. Em outras, geralmente equações de auto-valores, os operadores de fatorização são identificados como operadores de levantamento e de abaixamento de índices, de modo que basta resolver a equação diferencial correspondente ao auto-valor de nível mais baixo, onde uma simples integração é suficiente. Verifica-se que a maioria das equações diferenciais comumente estudadas na Física Matemática podem ser fatoradas.

We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities relating differential systems are obtained. They include a new identity relating Appell's hypergeometric system to a product of two Gauss' hypergeometric differential equations by a cubic transformation.