Monotonicity of zeros of Laguerre polynomials

Denote by x_nk(α), k=1,…,n, the zeros of the Laguerre polynomial . We establish monotonicity with respect to the parameter α of certain functions involving x_nk(α). As a consequence we obtain sharp upper bounds for the largest zero of .     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.     

Weak-type inequalities for higher order Riesz-Laguerre transforms

In this paper we study the weak-type (1,1) boundedness of the higher order Riesz-Laguerre transforms associated with the Laguerre polynomials. In particular, we obtain the boundedness for the Riesz-Laguerre transforms of order 2 and we find also the sharp polynomial weight ω that makes the Riesz-Laguerre transforms of order greater than two continuous from into L1,∞(dμα), being μα the Laguerre measure.     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved

     

A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs

We prove a theorem that for an integer s?0, if 12s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K_n, where n=(12s+7)(6s+7), is at least . By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least 2^αn?+o(n?), ?>1, nonisomorphic nonorientable triangular embeddings of K_n for n=6t+1, . To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given.     

Generating functions via integral transforms

In this paper, we use some integral transforms to derive, for a polynomial sequence {Pn(x)}_n?0, generating functions of the type , starting from a generating function of type , where {γn}_n?0 is a real numbers sequence independent on x and t. That allows us to unify the treatment of a generating function problem for many well-known polynomial sequences in the literature.     

A simple proof for a theorem of Luxemburg and Zaanen

In this paper a simple proof for the following theorem, due to Luxemburg and Zaanen is given: an Archimedean vector lattice A is Dedekind σ-complete if and only if A has the principal projection property and A is uniformly complete. As an application, we give a new and short proof for the following version of Freudenthal's spectral theorem: let A be a uniformly complete vector lattice with the principal projection property and let 0<u∈A. For any element w in A such that 0?w?u there exists a sequence in A which satisfies , where each element sn is of the form , with real numbers α₁,…,αk such that 0?αi?1 (i=1,…,k) and mutually disjoint components p₁,…,pk of u.     

Equality of norm groups of subextensions of Sn(n?5) extensions of algebraic number fields

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S_n on n?5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality to hold, where is the group of norms from K to k of the elements of the multiplicative group K∗ of K.     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

Strong convergence theorems for uniformly continuous pseudocontractive maps

Let K be a nonempty closed convex subset of a real Banach space E and let be a uniformly continuous pseudocontraction. Fix any u∈K. Let {xn} be defined by the iterative process: x₀∈K, xn+1:=μn(αnTxn+(1−αn)xn)+(1−μn)u. Let δ(?) denote the modulus of continuity of T with pseudo-inverse ?. If and {xn} are bounded then, under some mild conditions on the sequences n{αn} and n{μn}, the strong convergence of {xn} to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on and {xn} can be dispensed with.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Divisibility properties of generalized Laguerre polynomials

In this paper we give effective upper bounds for the degree k of divisors (over ?) of generalized Laguerre polynomials L^α_n(x), i.e. of for α = −tn − s − 1 and α = tn + s with t,s ∈ ?, t = O(log k), s = O(k log k) and k sufficiently large.     

Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each the nth Laguerre polynomial has an m-fold zero at the origin when α=−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers.     

Variation of parameters and solutions of composite products of linear differential equations

Given a basis of solutions to k ordinary linear differential equations ?j[y]=0(j=1,2,…,k), we show how Green's functions can be used to construct a basis of solutions to the homogeneous differential equation ?[y]=0, where ? is the composite product ?=?₁?₂…?k. The construction of these solutions is elementary and classical. In particular, we consider the special case when . Remarkably, in this case, if {y₁,y₂,…,yn} is a basis of ?₁[y]=0, then our method produces a basis of for any k∈N. We illustrate our results with several classical differential equations and their special function solutions.     

The Laplacian spectral radius of some bipartite graphs

In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K_1,k and one vertex of degree n of K_m,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs.