In a recent paper Ragab and Simary (6) have deduced some integrals involving the products of generalized Whittaker functions. We feel that some of the integrals are not correctly evaluated. Indeed the integral (2) should be symmetrical in α and β, as we see by changing the variables by s = − t in the integral, while it is not symmetrical. A similar remark will follow in the case of the integrals (3) and (6).

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

Let be the Jacobi polynomial as defined by Szegö in (7) (see equation (4) below.) Carlitz in (2) presented among others the following formula

Although, as Carlitz claims, this formula may be derived directly from the definition of Jacobi polynomials, a probabilistic proof such as presented below may shed some new light on formula (1), as well as suggest probabilistic proofs for other similar formulas of Jacobi polynomials, e.g. those given by Manocha and Sharma in (4) and (5) and by Manocha in (3). In addition, it is quite possible that this method of proof will result in the derivation of some new formulas for Jacobi polynomials.

1. For the familiar definition of (R, λn, κ), (R*, λn, κ) and (N, p) means and their notations, see, for example (3). If {fn} is any arbitrary sequence, we adopt the convention throughout that f−1 = 0. A method of absolute summability |A| is said to be ineffective if it is absolutely regular and sums only absolutely convergent sequences. The main object of this note is to prove absolute analogues of Kuttner's results in (3). And for this we need to prove a theorem (Theorem 1) on ineffectiveness for |N, p| method.

Slater ((5), p. 27) and Bose ((1), p. 202) have obtained certain recurrence relations about Whittaker and hypergeometric functions. In this note an attempt has been made to obtain generalizations of these recurrence relations.

The Gontcharoff polynomials Gn(z; α0, α1,…, αn − 1); n = 0, 1, 2,… are defined to be the polynomials orthogonal to the linear functionals Ln(f) = f(n)(αn), for a specified sequence of complex numbers {αn}. We will abbreviate Gn(z; α0, α1,…, αn−1) to Gn(z) when the sequence {αn} is not in doubt, so that

We let h1/k1,h2/k2…hN(n)/kN(n)represent the Farey series of order n. It is well known that

Recently, Gupta (l) proved that

and Hans and Chander (2) proved that

An identity for a rational function is used, in conjunction with generating functions for a certain class of polynomials, to derive infinite sets of identities for these polynomials. Identities are given for the polynomials of Legendre, Gegenbauer, Hermite, Tchebycheff and L. Carlitz. A novel type of classification for these polynomials is indicated. Next, infinite sequences of identities for various Bessel functions and for power functions are derived. New identities for trigonometric functions are also given, as well as brief miscellanea on: functions resembling Dirac's delta function, integral transforms and a functional equation.

A simple formula is derived for transforming the slowly convergent series involving Bessel functions which arise in axially symmetric potential problems into more rapidly convergent series. Using it, sums of series previously found by elaborate special methods may be obtained by direct substitution. The necessary numerical coefficients are given.

Cases in which heat is produced in solids are becoming increasingly important in technical applications ((3), pp. 11–12).

Non-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.

The authors, while attempting to give a new generalization of the Laplace transform, have come across a particular function in the form of Meijer's G-function

which, when taken as a nucleus of the transformation K(x) in

defines a new transform. This transform besides serving as a generalization of the Laplace transform and most of its generalizations existing in the literature bears the characteristic property of generalizing the Hankel transform (and hence some of its generalizations) as well. In this paper the authors after having defined the transform have given an inversion theorem which is supported by means of two examples.

We consider in this paper the solution behaviour as s → 0 of the equation

where the primes indicate differentiation with respect to s, and a, b, c are constants.

1. In (1) M. G. Smith solved the equation

where

for R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namely

and

The Stieltjes transform is a consequence of the iteration of two Laplace transforms, namely if

then is referred to as the Stieltjes transform of g(t).

In a long series of papers published some years ago both here and elsewhere, Dr L. J. Slater has derived a large number of valuable results concerned with the theory of generalized hypergeometric functions.

The long open question regarding the almost everywhere convergence of the Fourier series of an L2-function having been recently settled by Carleson (1), an interesting problem still awaits solution. It is to investigate whether the Fourier series of a function of class Lp (1 < p < 2), converges (or diverges) almost everywhere; for it is already known that the Fourier series of an integrable function may diverge almost everywhere. See Kolmogorov (3) and Marcinkiewicz (4) or ((8), 1, p. 305), see also Kahane(2). The object of this paper is simply to show that a suitable restriction imposed on the modulus of smoothness of an integrable function guarantees the almost everywhere convergence of its Fourier series. This will establish, in a more general form, the author's conjecture made in (5). Our proof is based on Carleson's theorem according to which the Fourier series of an L2-function converges almost everywhere.

In a simple homogeneous birth-and-death process with λ and μ as the constant birth and death rates respectively, let X(t) denote the population size at time t, Z(t) the number of deaths and N(t) the number of events (births and deaths combined) occurring during (0, t). Also let . The results obtained include the following:

(a) An explicit formula for the characteristic quasi-probability generating function of the joint distribution of X(t), Y(t) and Z(t).

(b) Let X(0) = 1. It is shown that, if t → ∞ while λ ≤ μ, N(t) ↑ N a.s., where N takes only positive odd integral values. If λ > μ, then P[N(t) ↑ ∞] = 1 − μ/λ. Given that N(t)∞, the limiting distribution of N(t) is similar to that of N. It was reported earlier (Puri (11)), that the limiting distribution of Y(t) is a weighted average of certain chi-square distributions. It is now found that these weights are nothing but the probabilities P[N = 2k + 1] (k = 0, 1,…).

(c) Let λ = μ, and MXω), MYω and MZω be defined as in (36), then as

where the c.f. of (X*; Y*; Z*) is given by (38).

(d) Exact expressions for the p.d.f. of Y(t) are derived for the cases (i) λ = 0, μ > 0, (ii) λ > 0, μ = 0. For the case (iii) λ gt; 0, μ > 0, since the complete expression is complicated, only the procedure of derivation is indicated.

(e) Finally, it is shown that the regressions of Y(t) and of Z(t) on X(t) are linear for X(t) ≥ 1.

The main object of this paper is to derive an expansion formula for a generalized hypergeometric function of two variables in a series of products of generalized hypergeometric functions of two variables and a Meijer's G-function. The result established in this paper is the extension of the results recently given by Srivastava (5) and Verma (6). It is interesting to note that some interesting expansions can be derived from the result by specializing the parameters.

Let a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the form

for large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.