On convergence of basic hypergeometric series (original) (raw)
Some expansions of hypergeometric functions in series of hypergeometric functions†
Glasgow Mathematical Journal, 1976
Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.
Dn Basic Hypergeometric Series
Ramanujan Journal, 1999
We study multiple series extensions of basic hypergeometric series related to the root system Dn. We make a small change in the notation used for Cn and Dn series to bring them closer to An series. This allows us to combine the three types of series, and get Dn extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' 6 φ5 sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define An and Cn extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system An (U(n + 1) basic hypergeometric series).
A Note on the 2F1 Hypergeometric Function
2010
The special case of the hypergeometric function 2F1_{2}F_{1}2F1 represents the binomial series (1+x)alpha=sumn=0infty(:alphan:)xn(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}(1+x)alpha=sumn=0infty(:alphan:)xn that always converges when ∣x∣<1|x|<1∣x∣<1. Convergence of the series at the endpoints, x=pm1x=\pm 1x=pm1, depends on the values of alpha\alphaalpha and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series 2F1(alpha,beta;beta;x)_{2}F_{1}(\alpha,\beta;\beta;x)2F1(alpha,beta;beta;x) for ∣x∣<1|x|<1∣x∣<1 and obtain new result on its convergence at point x=−1x=-1x=−1 for every integer alphaneq0\alpha\neq 0alphaneq0. The proof is within a new theoretical setting based on the new method for reorganizing the integers and on the regular method for summation of divergent series.
An extension of Pochhammer’s symbol and its application to hypergeometric functions, II
Filomat, 2018
Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.
An analytic method for convergence acceleration of certain hypergeometric series
Mathematics of Computation, 1995
A method is presented for convergence acceleration of the generalized hypergeometric series 3F2 with the argument ±1 , using analytic properties of their terms. Iterated transformation of the series is performed analytically, which results in obtaining new fast converging expansions for some special functions and mathematical constants.
An extension of Pochhammer’s symbol and its application to hypergeometric functions
Filomat, 2017
By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.
Quaderni di Matematica, 2004
Partitions and representations 1 A2. Ferrers diagrams of partitions 2 A3. Addition on partitions 6 A4. Multiplication on partitions 6 A5. Dominance partial ordering 7 Chapter B. Generating Functions of Partitions Il BI. Basic generating functions of partitions B2. Classical partitions and the Gauss formula B3. Partitions into distinct parts and the Euler formula 19 B4. Partitions and the Gauss q-binomial coefficients B5. Partitions into distinct parts and finite q-differences Chapter C . Durfee Rectangles and Classical Partition Identities Cl. q-Series identities of Cauchy and Kummer: Unification C2. q-Binomial convolutions and the Jacobi triple product C3. The finite form of Euler's pentagon number theorem Chapter D. The Carlitz Inversions and Rogers-Ramanujan Identities DI. Combinatorial inversions and series transformations D2. Finite q-differences and further transformation D3. Rogers-Ramanujan identities and their finite forms Chapter E. Basic Hypergeometric Series El. Introduction and notation E2. The q-Gauss summation formula E3. Transformations of Heine and Jackson E4. The q-Pfaff-Saalschiitz summation theorem E5 . The terminating q-Dougall-Dixon formula E6. The Sears balanced transformations E7. Watson's q-Whipple transformation vii viii CHU Wenchang and DI CLAUDIO Leontina Chapter F. Bilateral Basic Hypergeornetric Series Fl. Definition and notation F2. Rarnanuj an's bilateral -series identity F3. Bailey's bilateral identity F4. Bilateral q-analogue of Dixon's theorern F5. Partial fraction decornposition rnethod Chapter G. The Lagrange Four Square Theorern G 1. Representations by two square surns G2. Representations by four square surns G3. Representations by six square surns G4. Representations by eight square surns G5. Jacobi's identity and q-difference equations Chapter H. Congruence Properties of Partition Function Hl. Proof of p(5n + 4) = O (rnod 5) H2. Generating function for p(5n + 4) H3. Proof of p(7n + 5) O (rnod 7) H4. Generating function for p(7n + 6) H5. Proof of p(l1n + 6) O (rnod 11
Some hypergeometric and other evaluations of ζ(2) and allied series
Applied Mathematics and Computation, 1999
Numerous interesting solutions of the problem of evaluating the Riemann f2 X I k1 k À2 , which was of vital importance to Euler and the Bernoulli brothers (James and John Bernoulli), have appeared in the mathematical literature ever since Euler ®rst solved this problem in 1736. The main object of this note is to present yet another evaluation of f2 and related sums by applying the theory of hypergeometric series. Some other approaches to this problem are also indicated.
Hypergeometric Summation Revisited
Computer Algebra 2006: Latest Advances in Symbolic Algorithms - Proceedings of The Waterloo Workshop, 2007
We consider hypergeometric sequences, i.e., the sequences which satisfy linear rst-order homogeneous recurrence equations with relatively prime polynomial coe cients. Some results related to necessary and su cient conditions are discussed for validity of discrete Newton-Leibniz formula P w k=v t(k) = u(w + 1 ) ; u(v) w h e n u(k) = R(k)t(k) and R(k) is a rational solution of Gosper's equation.
On the summation of some divergent hypergeometric series and related perturbation expansions
1990
Divergent hypergeometric series 2 FO(a,p;-l/r) occur frequently in Poincare-type asymptotic expansions of special functions. These divergent series 2F0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied. There is considerable evidence that Levin's sequence transformation (1973) and in particular also some closely related sequence transformations (Weniger (1989)), which were derived recently, sum divergent series 2F0 much more efficiently than Pad6 approximants. Similar summation problems occur also in the case of divergent Rayleigl-Schrodinger perturbation expansions of elementary quantum mechanical systems. A comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that Levin's sequence transformation and the recently derived new sequence transformations are again more efficient than Pad& approximants. However, the superiority of Levin's sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion.
Summation of some infinite series by the methods of Hypergeometric functions and partial fractions
BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS
In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].
Journal of Mathematics Research, 2010
In the present paper, we obtain numerical values for Gaussian hypergeometric summation theorems by giving particular values to the parameters a, b and the argument x; three summation theorems for 2 F 3 (1 4 , 3 4 ; 1 2 , 1 2 , 1; x), three summation theorems for 4 F 3 (1 2 , 1 2 , 1 2 , a+b b ; 1, 1, a b ; x), two summation theorems for 4 F 3 (1 2 , 1 3 , 2 3 , a+b b ; 1, 1, a b ; x), four summation theorems for 4 F 3 (1 2 , 1 6 , 5 6 , a+b b ; 1, 1, a b ; x) and ten summation theorems for 4 F 3 (1 2 , 1 4 , 3 4 , a+b b ; 1, 1, a b ; x).
Contiguous relations of hypergeometric series
Journal of Computational and Applied Mathematics, 2003
The 15 Gauss contiguous relations for 2F1 hypergeometric series imply that any three 2F1 series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them.
CERTAIN TRANSFORMATION FORMULAE FOR BASIC HYPERGEOMETRIC SERIES
Gasper (1) established the q-analogue of Karlsson-Minton summation formula in the form : q, b ,......., b a q , b, b q ,......, b q ; q; 1 r 1-(m m ...... m ) m r m 2 1 1 2 r 1 r 1 b a r r 1 2 r 1 1 m m ...... m 1 1 , ; ; ........... ; , ; ; ........ ; b q b q b q a q bq q b b q b b q a bq a r r m r m m r m (1.1) where m1, m2,......, mr are non-negative integers A particular case of the above summation formula is the following one : , , , , , ; ; 1 2 1 ( ) 1 2 4 3 q q q q q m n m n n m n