Q-Analogue of a Two Variable Inverse Pair of Series with Applications to Basic Double Hypergeometric Series

1989 ◽  
Vol 41 (4) ◽  
pp. 743-768 ◽  
Author(s):  
Christian Krattenthaler
Keyword(s):  
Hypergeometric Series ◽  
Formal Series ◽  
Unique Pair ◽  
Image Position

Let be a pair of a formal series (fps) in z1 and z2 of the form where is an fps with for i = 1,2. Then there exists a unique pair of fps which is also of the form (1.1), with This pair is called the inverse of f(z1,z2 ).

Certain Summation Formulae for Basic Hypergeometric Series

1977 ◽  
Vol 20 (3) ◽  
pp. 369-375 ◽  
Author(s):  
Arun Verma
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Summation Formulae ◽  
Image Position

In 1927, Jackson [5] obtained a transformation connecting awhere N is any integer, with aviz.,1where | q | > l and |qγ-α-βN| > l.

Special values of the hypergeometric series

1991 ◽  
Vol 109 (2) ◽  
pp. 257-261 ◽  
Author(s):  
G. S. Joyce ◽  
I. J. Zucker
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Special Values ◽  
Image Position

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such asand where Γ(x) denotes the gamma function.

scholarly journals Fatou-Riesz-theorems in general sequence spaces

1980 ◽  
Vol 23 (2) ◽  
pp. 199-205
Author(s):  
Gerhard Otto Müller ◽  
Rolf Trautner
Keyword(s):  
Power Series ◽  
Formal Series ◽  
Sequence Spaces ◽  
Partial Sums ◽  
Classical Theorem ◽  
General Sequence ◽  
Image Position

Consider a formal serieswith partial sumsand the corresponding power series. Throughout we will assume thatfis analytic for |z| <1, i.e. thatA classical theorem of Fatou-Riesz (see (1,4)) states that ifandthenis convergent to 0.

Zeros of generalized hypergeometric functions

1973 ◽  
Vol 74 (2) ◽  
pp. 269-276
Author(s):  
A. D'Adda ◽  
R. D'Auria
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
Image Position

In this paper we derive the conditions which have to be satisfied in order to obtain some classes of zeros of the generalized hypergeometric series of the typeThese conditions read:

Periods and special values of the hypergeometric series

1989 ◽  
Vol 106 (3) ◽  
pp. 389-401 ◽  
Author(s):  
Matthias Flach
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Joint Paper ◽  
Special Values ◽  
Precise Statement ◽  
Image Position

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities likebut it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

Properties (V) and (w V) on C(Ω X)

1995 ◽  
Vol 117 (3) ◽  
pp. 469-477 ◽  
Author(s):  
Elizabeth M. Bator ◽  
Paul W. Lewis
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Linear Functional ◽  
Formal Series ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Weakly Compact ◽  
Image Position ◽  
Continuous Function Space ◽  
Fundamental Paper

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

An Inclusion Relation for Abel, Borel, and Lambert Summability

1980 ◽  
Vol 32 (3) ◽  
pp. 695-702
Author(s):  
W. Gawronski ◽  
H. Siebert ◽  
R. Trautner
Keyword(s):  
Formal Series ◽  
Partial Sums ◽  
Inclusion Relation ◽  
Inclusion Theorem ◽  
New Type ◽  
Image Position

In this paper a new type of inclusion theorem concerning Abel, Borel and Lambert summability is established. To state our results we need some definitions and notations. With a formal series ∑k=0∞ak, ak∈ C, and its partial sums snwe associate the seriesThen ∑k=0∞ ak is said to be summable to the value s(a) by Abel's method, if (1.1) is convergent for |v| > 1 and limv→1+A(v)= s,(b) by Lambert's method, if (1.2) is convergent for |v| > 1 and limv→1+L(v)= s,(c) by Borel's method, if (1.3) is convergent for all x ∈ R and limx→+∞B(x)= s,

On the partial sums of series of hypergeometric type

1953 ◽  
Vol 49 (3) ◽  
pp. 441-445 ◽  
Author(s):  
R. P. Agarwal
Keyword(s):  
Infinite Series ◽  
Hypergeometric Series ◽  
Partial Sums ◽  
Hypergeometric Type ◽  
The Subject ◽  
Image Position

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

Probabilistic factorization of a quadratic matrix polynomial

1990 ◽  
Vol 107 (3) ◽  
pp. 591-600 ◽  
Author(s):  
Joanne Kennedy ◽  
David Williams
Keyword(s):  
Matrix Polynomial ◽  
Unique Pair ◽  
Finite Set ◽  
Symmetric Element ◽  
Algebraic Result ◽  
Quadratic Matrix ◽  
Image Position

A purely algebraic result. We begin by stating the following theorem. Theorem. Let E be a finite set, and letdenote the set of real E × E matrices with non-negative off-diagonal elements and with non-positive row sums. Let A be a symmetric element of, and let V be a diagonal real E × E matrix. Then there exists a unique pair (H+, H−) of elements ofsuch thatI denoting the identity E × E matrix, and the superscript T signifying transpose. It is an immediate consequence that

On the converse of Mertens' theorem

1973 ◽  
Vol 73 (3) ◽  
pp. 467-471 ◽  
Author(s):  
K. A. Jukes
Keyword(s):  
Formal Series ◽  
Cauchy Product ◽  
Image Position ◽  
Dirichlet Product

Let (λm), (µn) (m, n = 0, 1, 2,…) satisfyrespectively. Let vp (p = 0, 1, 2, …) be the sequence (λm+µn) arranged in ascending order, equal sums λm+µn being considered as giving just one vp Then for given formal series Σam, Σbn the formal series C = Σ cp whereis called the general Dirichlet product of Σamand Σbn (see Hardy (2), p. 239). When λn = µn = n we have the Cauchy product. In the case λn = logm, µn = logn (m, n = 1, 2,…) we have vp =log p(p = 1, 2, …)and it is natural to call C the ordinary Dirichlet product.

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