On the zeros of d-symmetric d-orthogonal polynomials

In this paper, we give some properties of the zeros of d-symmetric d-orthogonal polynomials and we localize these zeros on (d+1) rays emanating from the origin. We apply the obtained results to some known polynomials. In particular, we partially solve the conjecture about the zeros of the Humbert polynomials stated by Milovanovi? and Dordevi? [G.V. Milovanovi?, G.B. Dordevi?, On some properties of Humbert's polynomials, II, Ser. Math. Inform. 6 (1991) 23-30]. A study of the eigenvalues of a particular banded Hessenberg matrix is done.     

d-Symmetric d-orthogonal polynomials of Brenke type

In this paper, we characterize the d-symmetric d-orthogonal polynomials of Brenke type. We obtain two new families of polynomials and the moments of the corresponding d-dimensional vectors of linear functionals. Weights, providing integral representations for these moments, were also given.     

Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam-Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson-Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.     

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.     

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big q-Jacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.     

Approximation properties for generalized q-Bernstein polynomials

In this paper, we introduce the generalized q-Bernstein polynomials based on the q-integers and we study approximation properties of these operators. In special case, we obtain Stancu operators or Phillips polynomials.     

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.     

Notes on generalization of the Bernoulli type polynomials

Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.     

A q-summation formula,the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities.     

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.     

Some identities and congruences concerning Euler numbers and polynomials

In this paper, using the properties of the moments of p-adic measures, we establish some identities and Kummer likewise congruences concerning Euler numbers and polynomials. In the preliminaries, we introduce the Laplace transform which is an important tool for the determination of the moments of p-adic measures. We also give a sequence n(dn) linked to Euler numbers and which satisfies the same type of congruences and identities as the Euler numbers. At the end, for p=2, we give congruences on Euler numbers involving the sequence n(dn).     

A curious q-analogue of Hermite polynomials

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).     

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.     

Markov-Bernstein inequalities for generalized Gegenbauer weight

The Markov-Bernstein inequalities for generalized Gegenbauer weight are studied. A special basis of the vector space Pn of real polynomials in one variable of degree at most equal to n is proposed. It is produced by quasi-orthogonal polynomials with respect to this generalized Gegenbauer measure. Thanks to this basis the problem to find the Markov-Bernstein constant is separated in two eigenvalue problems. The first has a classical form and we are able to give lower and upper bounds of the Markov-Bernstein constant by using the Newton method and the classical qd algorithm applied to a sequence of orthogonal polynomials. The second is a generalized eigenvalue problem with a five diagonal matrix and a tridiagonal matrix. A lower bound is obtained by using the Newton method applied to the six term recurrence relation produced by the expansion of the characteristic determinant. The asymptotic behavior of an upper bound is studied. Finally, the asymptotic behavior of the Markov-Bernstein constant is O(n²) in both cases.     

Reliability evaluation of multi-state systems under cost consideration

A more practical and desirable performance index of multi-state systems is the two-terminal reliability for level (d, c) (2TR_d,c), defined as the probability that d units of flow can be transmitted from the source node to the sink node with the total cost less than or equal to c. In this article, a simple algorithm is developed to calculate 2TR_d,c in terms of (d, c)-MPs. Two major advantages of the proposed algorithm include: (1) as of now, it is the only algorithm that searches for (d, c)-MPs without requiring all minimal paths (MPs) and the procedure of transforming feasible solutions; (2) it is more practical and efficient in solving (d, c)-MP problem in contrast to the best-known method. An example is provided to illustrate the generation of (d, c)-MPs by using the presented algorithm, and 2TR_d,c is thus evaluated. Furthermore, the computational experiments are conducted to verify the performance of the presented algorithm.     

Second structure relation for q-semiclassical polynomials of the Hahn Tableau

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.     

On the Krall-type discrete polynomials

In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.     

Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball.