THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

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Handbook of Hypergeometric Integrals

Special Functions For Scientists And Engineers

1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series 9. Basic Appell Series.

Generalized Hypergeometric Functions. By Lucy Joan Slater. Pp. xiii, 273. 70s. 1966. (Cambridge University Press.)

The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kampe de Feriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol type numbers and polynomials. Using Euler's formula, relations among special functions, Hermite-type polynomials, the Chebyshev polynomials and the Dickson polynomials are given. Using generating functions and their functional equations, various formulas and identities are given. With help of computational formula for new families of special polynomials, some of their numerical values are given. Using hypegeometric series, trigonometric functions and the Euler's formula, some applications related to Hermite-type polynomials are presented. Finally, further remarks, observations and comments about generating functions for new families of special polynomials are given.

Some classes of generating functions for generalized Hermite- and Chebyshev-type polynomials: Analysis of Euler's formula

In this paper, we obtain two Gaussian hypergeometric summation formulas and show how each of these summation formulas can be applied to derive several general double-series identities and various generalized hypergeometric representations of such combinations of the exponential and trigonometric functions as The results presented here are presumably new.

Some hypergeometric summation formulas and series identities associated with exponential and trigonometric functions

A family of hypergeometric integrals associated with Ramanujan's integral formula

The main object of this paper is to derive a number of general double series identities and to apply each of these identities in order to deduce several hypergeometric reduction formulas for the Srivastava-Daoust double hypergeometric function. The results presented in this paper are based essentially upon some known hypergeometric summation theorems in conjunction with series rearrangement techniques. Mathematics Subject Classication 2000 : Primary 33C20; Secondary 33C70.

Applications of Some Hypergeometric Summation Theorems Involving Double Series

Some Applications of Celebrated Master Theorem of Ramanujan

The main aim of this paper is to develop some interesting applications, extensions and generalizations of Ramanujan's integral (which he obtained with the help of the celebrated Ramanujan Master theorem). Motivated essentially by some recent works by M. Garg and S. Mittal (see for example, [South East Asian J. Math. Math. Sc., 1(2) (2003), 85-95; Proc. Indian Acad. Sci. (Math. Sci.), 114(2) (2004), 99-101; Kyungpook Math. J., 44 (2004), 473-479]) and H. M. Srivastava et al. (see for example, [Rev. Roumaine Phys., 32 (1987), 685-692; Advanced Studies in Contemporary Mathematics, 18(2) (2009), 113-125]), we evaluate several general families of new definite integrals involving functions of one and more complex variables, some of which are then applied to derive the corresponding results associated with generalized Gaussian hypergeometric function qpF and the Fox-Wright generalized hypergeometric functions .

M. I. Qureshi

Papers

Some hypergeometric summation formulas and series identities associated with exponential and trigonometric functions

A family of hypergeometric integrals associated with Ramanujan's integral formula

Applications of Some Hypergeometric Summation Theorems Involving Double Series

Some Applications of Celebrated Master Theorem of Ramanujan

A class of hypergeometric generalizations of an integral of srinivasa ramanujan