Distributional solutions of the hypergeometric differential equation

We present the distributional solutions to the hypergeometric differential equation. These solutions are obtained in the form of infinite series of the Dirac Delta functions and its derivatives. We employ these solutions to observe their interesting features. Furthermore, the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials.     

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.     

On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. (SIAM J. Math. Anal. 10 (3) (1979) 655).     

Hyperasymptotic expansions of confluent hypergeometric functions

Hyperasymptotic expansions were recently introduced by Berryand Howls, and yield refined information by expanding remaindersin asymptotic expansions. This paper gives a new method forobtaining hyperasymptotic expansions for integrals representingthe confluent hypergeometric U-function. At each level, theremainder is exponentially small compared with the previousremainders, and the number of new terms is increasing. Threenumerical illustrations confirm these exponential improvements.     

On some particular solutions of the wave equation

We consider solutions of the wave equation in the form of functions that describe waves propagating at velocities different from the constant occurring in the equation. We present new dependences and give an interpretation for them.     

On the expansion of confluent hypergeometric functions in terms of Bessel functions

For the confluent hypergeometric functions U (a, b, z) and M (a, b, z) asymptotic expansions are given for a → ∞. The expansions contain modified Bessel functions. For real values of the parameters rigorous error bounds are given.     

Properties of the zeros of confluent hypergeometric functions

Several infinite systems of nonlinear algebraic equations satisfied by the zeros of confluent hypergeometric functions are derived. Certain sum rules and other related properties for the zeros follow from these equations. A large class of special functions, which are special cases of confluent hypergeometric functions, is included. This is illustrated in the case of the zeros of Bessel functions and Laguerre polynomials.     

Exact and explicit solutions for a nonlinear extended iterative differential equation

This paper is concerned with a nonlinear iterative functional differential equation x′(z) = 1/x(p(z) + bx′(z)). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.     

On some solutions to the Chapman-Kolmogorov integral equation

By applying integral transformations, we obtain some solutions to the Chapman-Kolmogorov equation. These are illustrated by examples.     

Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T: E → X for which dens X/E = dens X/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c ₀ is automorphic and conjectured that c ₀ and ℓ₂ are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c ₀(Γ), for Γ uncountable. That c ₀(Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of ℓ _n ^p , 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as L _p (μ), p ≠ 2, C(K) spaces not isomorphic to c ₀ for K metric compact, subspaces of c ₀ which are not isomorphic to c ₀, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic. The work of the first author has been supported in part by project MTM2004-02635     

一类复系数微分方程解的增长性

研究了一类系数是亚纯函数的高阶微分方程解的性质,假设其中某一个系数具有有限亏值,然后对其它的系数添加相应的限制条件,使得方程的每一个非零亚纯解都具有无穷级.     

The structure of solutions of a hypergeometric equation system

We consider a system of ordinary differential equations which is a multidimensional analogue of a hypergeometric equation. We study the structure and asymptotics of solutions at the singular points and construct a fundamental system of solutions in a neighborhood of each singular point. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 52–64, January–March, 1999. Translated by V. Mackevičius     

On convergence of the solutions of a functional differential equation

A sufficient condition is given for the solutions of a functionally perturbed linear system of ordinary differential equations to have limits at ± ∞.     

On hypergeometric type partial differential equations

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 1, pp. 82–90, January–March, 1995.