A nonterminating 8φ7 summation for the root system Cr

Using multiple q-integrals and a determinant evaluation, we establish a nonterminating ₈φ₇ summation for the root system C_r. We also give some important specializations explicitly.     

Extensions of certain classical integrals of Erdélyi for Gauss hypergeometric functions

It is shown how series manipulation technique and certain classical summation theorems for hypergeometric series can be used to prove Erdélyi's integral representations for ₂F₁(z), originally proved using fractional calculus. The method not only leads to generalizations but also leads to new integrals of Erdélyi type for certain _q+1F_q(z) and corresponding Pochhammer contour integrals. The technique outlined here, compared to the method of fractional calculus, seems to be more effective as it not only provides transparent elementary proofs of Erdélyi's integrals but even leads to various generalizations.     

An extremal problem on potentially Kr,s-graphic sequences

We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ(K_r,s,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(K_r,s,n) is potentially K_r,s-graphic, where K_r,s is a r×s complete bipartite graph, i.e., π has a realization G containing K_r,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially K_r,s-graphic, and then we determine σ(K_r,r,n) for r=3,4.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

Some New Applications of Matrix Inversions in A r

We apply multidimensional matrix inversions to multiple basic hypergeometric summation theorems to derive several multiple (q-)series identities which themselves do not belong to the hierarchy of (basic) hypergeometric series. Among these are A terminating and nonterminating q-Abel and q-Rothe summations. Furthermore, we derive some identities of another type which appear to be new already in the one-dimensional case.     

Elementary derivations of identities for bilateral basic hypergeometric series

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For motivation, we review our previous simple proof (Proc. Amer. Math. Soc. 130 (2002), 1103-1111) of Bailey's very-well-poised ₆y₆_6\psi_6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised ₈y₈_8\psi_8 transformation, a very-well-poised ₁₀y₁₀_{10}\psi_{10} transformation, by induction, Slater's general transformation for very-well-poised _2ry_2r_{2r}\psi_{2r} series, and Slater's transformation for general _ry_r_{r}\psi_{r} series. Finally, we derive some new transformations for bilateral basic hypergeometric series of a specific type.     

On generalized Boolean functions II

This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:Bⁿ→B can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBF_n[A]. In this paper the following questions are presented: (1) What is the relationship between GBF_n[A₁] and GBF_n[A₂] when A₁A₂. (2) What can be said about GBF_n[A₁∩A₂] and GBF_n[A₁A₂] in comparison with GBF_n[A₁]∩GBF_n[A₂] and GBF_n[A₁]GBF_n[A₂], respectively.     

Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ₁. We consider the power method, i.e. that of choosing a vector v₀ and setting v_k = A^kv₀; then the Rayleigh quotients R_k = (Av_k, v_k)/(v_k, v_k) usually converge to λ₁ as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close R_k is to λ₁. They are both based on a bound on λ₁ − R_k involving the difference of two consecutive Rayleigh quotients and a quantity ω_k. While we do not know how to directly calculate ω_k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ₁ − R_k which is proportional to (λ₂/λ₁)^2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ).     

ARCH-type bilinear models with double long memory

We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation X_t=ζ_tA_t+B_t,(), where are standard i.i.d. r.v.'s, and A_t,B_t are moving averages in X_s, s<t. Stationary solution of () is obtained as an orthogonal Volterra expansion. In the case A_t≡1, X_t is the classical AR(∞) process, while B_t≡0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, X_t may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters and , respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of X_t and X_t² and the asymptotic distribution of the corresponding partial sums’ processes.     

On the area of the intersection of disks in the plane

Let C₁,…, C_n and C′₁,…, C′_n be two collections of equal disks in the plane, with centers c₁,…, c_n and c′₁,…, c′_n. According to a well-known conjecture of Klee and Wagon (1991), if |c_i − c_j| ≥ |c′_i − c′_j| for all i, j, then Area(∩_i C_i) ≤ Area(∩_i C′_i).

We prove this statement in the special case when there is a continuous contraction of {c₁,…, c_n} onto {c′₁,…, c′_n}.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

Dn Basic Hypergeometric Series

We study multiple series extensions of basic hypergeometric series related to the root system D_n. We make a small change in the notation used for C_n and D_n series to bring them closer to A_n series. This allows us to combine the three types of series, and get D_n extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' _{6 5} 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 A_n and C_n 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 A_n (U(n + 1) basic hypergeometric series).     

Summation and transformation formulas for elliptic hypergeometric series

Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.     

Loopless generation of up–down permutations

An up–down permutation P=(p₁,p₂,…,p_n) is a permutation of the integers 1 to n which satisfies constraints specified by a sequence C=(c₁,c₂,…,c_n−1) of U's and D's of length n−1. If c_i is U then p_i<p_i+1 otherwise p_i−1>p_i. A loopless algorithm is developed for generating all the up–down permutations satisfying any sequence C. Ranking and unranking algorithms are discussed.     

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form

where W is a classical Weyl group, X is a one-dimensional character of W, and d(π) is the descent statistic. This completes a picture which is known when W is the symmetric group S_n (the Weyl group A_n−1). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B_n(C_n) and D_n. The B_n, case uses sign-reversing involutions, while the D_n case follows from a result of independent interest relating statistics for all three groups.     

Applications of hypergeometric summation theorems of kummer and dixon involving double series

Using series iteration techniques, we derive a number of general double series identities and apply each of these identities in order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.     

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.     

Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series

We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.     

Invariant record processes and applications to best choice modelling

Let X₁, X₂,…be identically distributed random variables from an unknown continuous distribution. Further let I_r(1), I_r(2),…be a sequence of indicator functions defined on X₁, X₂,…by I_r(k) = 0 if k < r, I_r(k) = 1 if X_k is a r-record AND = 0 otherwise. Suppose that we observe X₁, X₂,… at times T₁ < T₂ <… where the T_k's are realisations of some regular counting process (N(τ)) defined on the positive half-line. Having observed [0, τ], say, the problem is to predict the future behaviour of the counting processes (R_r(τ, s))_sτ = # r-records in [τ, s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N(τ))_τ0 has dependent increments.

The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.