On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

Weighted Least Square Convergence of Lagrange Interpolation on the Unit Circle

In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)²=P ^(a,β) _n(x),a>0,β>0,(1-x)P^(a,β) _n(x),a>0,β>-1,(1+x)P P^(a,β) _n(x),a>-1,β0 and P^(a,β) _n(x),a>-1,β>-1, respectively, onto the unit circle, where P^(a,β) _n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.     

Note on the Dočev-Grosswald asymptotic series for generalized Bessel polynomials

The Do?ev-Grosswald asymptotic series for the generalized Bessel polynomials y_n(z; a, b) is extended to O(1/n⁴) relative accuracy. The differential equation of the asymptotic factor, derived from the differential equation for y_n(z; a, b), is the basis of a different and easier method that employs simple recurrence relations and much less algebra for obtaining the same series. This is applied to the important special case of a = 1 to obtain the asymptotic series to O(1/n¹¹) relative accuracy.     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

On the mean square of the error term for the asymmetric two-dimensional divisor problems(II)

Let Δ(a, b; x) denote the error term of the asymmetric two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value ?_{n ￡ T} D²(a,b;n){\sum_{n\leq T} \Delta^2(a,b;n)} and the continuous mean value ò₁^TD²(a,b;x)dx{\int_1^T\Delta^2(a,b;x)dx} .     

A polynomial dual of partitions

Let P(x) = Σ_i=0ⁿa_ixⁱ be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where a_i of these points have degree i for 0≤i≤n. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G₁ × … × G_k having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Estimating the parameters of a truncated normal distribution

Let ø(x) be a truncated normal pdf over the interval [a,b], that is, assume ø(x)=exp[-(x–μ)²/2σ²]/∝^b_aexp[-(x–μ)²/2σ²]dx for - ∞<a?x?b?< + ∞ and zero elsewhere. Suppose that X₁,X₂,…,X_n is a random sample of size n from this truncated distribution. Using known properties of exponential families of distributions and the system of Legendre polynomials over the interval [-1,1], we examine the maximum likelihood estimation of the parameters μ and σ².     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.     

A binary linear recurrence sequence of composite numbers

Let (a,b)∈Z², where b≠0 and (a,b)≠(±2,−1). We prove that then there exist two positive relatively prime composite integers x₁, x₂ such that the sequence given by xn+1=axn+bxn−1, n=2,3,… , consists of composite terms only, i.e., |xn| is a composite integer for each n∈N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a,±1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a,b)=(1,1).     

A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E _n , Euler polynomials E _n (x), generalized Euler numbers E _n (a, b), generalized Euler polynomials E _n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E _n , E _n (x), E _n (a, b), E _n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.     

Dirichlet series associated with polynomials and applications

We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=_∑P(n₁,…,nk)?xΛ(n₁)?Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

The general divisor problem

Let Δ_k(x) = Δ(a₁, …, a_k; x) be the error term in the asymptotic formula for the summatory function of the general divisor function d(a₁, …, a_k; n), which is generated by ζ(a₁s) … ζ(a_ks) (1 ≤ a₁ ≤ … ≤ a_k are fixed integers). Precise omega results for the mean square integral ∫₁^x Δ_k²(x) dx are given, with applications to some specific divisor problems.     

A Monotonicity Property of Partial Orders

A proof using the FKG inequalities of the following result is obtained. Let P be a partially ordered set on a₁ ? a₂ ? ? ? a_m and b₁ ? b₂ ? ? ? b_n. Let P(x) be the proportion of linear extentions of P for which x holds. If x and y are disjunctions of conjunctions of additional inequalities of the form a_i ? b_j, then P(x and y) ? P(x)P(y). An example is provided that shows the result can be false if we don't assume the {a_i} and {b_j} are linearly ordered in P.     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

Linear maps on matrices: the invariance of symmetric polynomials

Let L be a linear map on the space M_n of all n by n complex matrices. Let h(x₁,…,x_n) be a symmetric polynomial. If X is a matrix in M_n with eigenvalues λ₁,…,λ_n, denote h(λ₁,…,λ_n) by h(X). For a large class of polynomials h, we determine the structure of the linear maps L for which h(X)=h(L(X)), for all X in M_n.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.