Showing papers on "Wilson polynomials published in 2011"
TL;DR: In this paper, infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems, where each polynomial has another integer n which counts the nodes and the totality of the integer indices correspond to the degrees of the virtual state wavefunctions which are deleted by the generalisation of Crum-Adler theorem.
214 citations
01 Jan 2011
TL;DR: In this article, the authors introduce and investigate some of the principal generalizations and unifications of each of these polynomials by means of suitable generating functions and present several interesting properties of these general polynomial systems including some explicit series representations in terms of the Hurwitz (or generalized) zeta function and the familiar Gauss hypergeometric function.
Abstract: In the vast literature in Analytic Number Theory, one can find systematic and extensive investigations not only of the classical Bernoulli, Euler and Genocchi polynomials and their corresponding numbers, but also of their many generalizations and basic (or q-) extensions. Our main object in this presentation is to introduce and investigate some of the principal generalizations and unifications of each of these polynomials by means of suitable generating functions. We present several interesting properties of these general polynomial systems including some explicit series representations in terms of the Hurwitz (or generalized) zeta function and the familiar Gauss hypergeometric function. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called -Stirling numbers of the second kind, we derive several properties and formulas and consider some of their interesting applications to the family of the Apostol type polynomials. We also give a brief expository and historial account of the various basic (or q-) extensions of the classical Bernoulli polynomials and numbers, the classical Euler polynomials and numbers, the classical Genocchi polynomials and numbers, and also of their such generalizations as (for example) the above-mentioned families of the Apostol type polynomials and numbers. Relevant connections of the definitions and results presented here with those in earlier as well as forthcoming investigations will be indicated.
180 citations
TL;DR: This paper investigates an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol–Genocchi hailing from the family of the Apostol type polynomes, and derives some basic properties and formulas and considers some interesting applications to the family.
162 citations
TL;DR: In this article, the authors give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials, and obtain some interesting identities on the qBernoullians.
Abstract: In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.
118 citations
TL;DR: Using a lemma proved by Akbary, Ghioca, and Wang, several theorems on permutation polynomials over finite fields are derived, which give not only a unified treatment of some earlier constructions of permutation Polynomial, but also new specific permutations polynOMials over F q .
114 citations
TL;DR: In this paper, the authors considered weighted ǫ-Genocchi numbers and polynomials and investigated some interesting properties of the weighted Ã-Gonzalez numbers related to weighted Ò-Bernstein polynomial by using fermionic à -adic integrals on Ω(ǫ).
Abstract: We consider weighted 𝑞-Genocchi numbers and polynomials. We investigated some interesting properties of the weighted 𝑞-Genocchi numbers related to weighted 𝑞-Bernstein polynomials by using fermionic 𝑝-adic integrals on ℤ𝑝.
89 citations
TL;DR: In this paper, the authors investigate special generalized Bernoulli polynomials that generalize classical Bernoullians and numbers, and prove a collection of extremely important and fundamental identities satisfied by their poly-Bernoullian and numbers.
Abstract: In this paper we investigate special generalized Bernoulli polynomials that generalize classical Bernoulli polynomials and numbers. We call them poly-Bernoulli polynomials. We prove a collection of extremely important and fundamental identities satisfied by our poly-Bernoulli polynomials and numbers. These properties are of arithmetical nature.
82 citations
TL;DR: It is shown how the Christoffel–Darboux formula for multiple orthogonal polynomials can be obtained easily using this information and gives explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynmials.
81 citations
TL;DR: In this article, the main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function and derive explicit representations for them in terms of a generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function.
Abstract: The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another
66 citations
TL;DR: In this article, it was shown that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closed at order 6 (as differential operators).
Abstract: We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.
57 citations
TL;DR: This paper obtains the explicit representation of this unified family of polynomials, in terms of a Gaussian hypergeometric function, as well as some symmetry identities and multiplication formula.
Abstract: In this paper, we present a unified family of polynomials including not only the Apostol-Bernoulli, Euler and Genocchi polynomials, but also a general family of polynomials suggested by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (10) (2010) 2779-2787]. We obtain the explicit representation of this unified family, in terms of a Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given.
TL;DR: This paper proves several symmetry identities for these generalized Apostol type polynomials by using their generating functions and derives several relations between the Apostol types, the generalized sum of integer powers and the generalized alternating sum.
Abstract: A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Luo and Srivastava [Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702-5728]. In this paper, we prove several symmetry identities for these generalized Apostol type polynomials by using their generating functions. As special cases and consequences of our results, we obtain the corresponding symmetry identities for the Apostol-Euler polynomials of higher order, the Apostol-Bernoulli polynomials of higher order and the Apostol-Genocchi polynomials of higher order, and also for another family of generalized Apostol type polynomials which were investigated systematically by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010) 2779-2787]. We also derive several relations between the Apostol type polynomials, the generalized sum of integer powers and the generalized alternating sum. It is shown how each of these results would extend the corresponding known identities.
TL;DR: In this article, the exceptional q-Racah polynomials were constructed, together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomorphisms.
Abstract: The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal polynomials of a single variable.
TL;DR: In this article, an extension of the nonterminating summation using the method of q-exponential operator and difference equation is given, which allows us to give a new generating function of the Askey-Wilson polynomials.
Abstract: In this paper, we give an extension of the non-terminating summation using the method of q-exponential operator and difference equation. This extended formula allows us to give a new generating function of the Askey–Wilson polynomials.
TL;DR: In this article, the Smith form of regular and singular $T$-palindromic matrix polynomials is analyzed over arbitrary fields. But the Smith forms are not applicable to the case of linear polynomial matrices.
Abstract: Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry,
a structure we call palindromic.
Several properties of scalar palindromic polynomials
are derived,
and together with properties of compound matrices,
used to establish the Smith form of regular and singular $T$-palindromic matrix polynomials
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials
are characterized,
and necessary conditions for the existence
of structured linearizations established.
In the odd degree case, a constructive procedure
for building palindromic linearizations
shows that the necessary conditions
are sufficient as well.
The Smith form for $*$-palindromic polynomials
is also analyzed.
Finally, results for palindromic matrix polynomials
over fields of characteristic two are presented.
TL;DR: Srivastava as mentioned in this paper gave an extension of the Bernoulii and Euler polynomials to the Stirling numbers of the second kind and derived several other formulas in series of Carlitz's $q$-Stirling numbers.
Abstract: The main object of this paper is to give $q$-extensions of several explicit relationships of H. M. Srivastava and \'{A}. Pint\'{e}r [\textit{Appl. Math. Lett.} {\bf 17} (2004), 375-380] between the Bernoulii and Euler polynomials. We also derive several other formulas in series of Carlitz's $q$-Stirling numbers of the second kind.
TL;DR: In this paper, the unification and generalization of three families of generalized Apostol type polynomials, which are ApostolBernoulli, Apostol-Euler, and Apostol Genocchi polynomorphisms, were studied.
Abstract: The aim of this paper is to investigate and introduce some new identities related to the unification and generalization of the three families of generalized Apostol type polynomials, which are Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, on the modern theory of the Umbral calculus and algebra. We also introduce some operators. Recently, Ozden constructed generating function of the unification of the Apostol type polynomials (see Ozden [H. Ozden, AIP Conf. Proc. 1281, (2010), 1125-1227.]). By using this generating function, we derive many properties of these polynomials. We give relations between these polynomials and Stirling numbers.
Journal Article•
TL;DR: In this article, the Euler-Seidel matrix method was used to obtain properties of geometric and exponential polynomials and numbers and some new results were obtained and some known results were reproved.
Abstract: In this paper we use the Euler-Seidel matrix method to obtain some properties of geometric and exponential polynomials and numbers. Some new results are obtained and some known results are reproved.
TL;DR: New algorithms for the nonclassic Adomian polynomials are presented, which are valuable for solving a wide range of nonlinear functional equations by theAdomian decomposition method, and their symbolic implementation in MATHEMATICA is introduced.
Abstract: In this article, we present new algorithms for the nonclassic Adomian polynomials, which are valuable for solving a wide range of nonlinear functional equations by the Adomian decomposition method, and introduce their symbolic implementation in MATHEMATICA. Beginning with Rach's new definition of the Adomian polynomials, we derive the explicit expression for each class of the Adomian polynomials, e.g. A"[email protected]?"k"="1^mf^(^k^)(u"0)Z"m","k for the Class II, III and IV Adomian polynomials, where the Z"m","k are called the reduced polynomials. These expressions provide a basis for developing improved algorithmic approaches. By introducing the index vectors, the recurrence algorithms for the reduced polynomials are suitably deduced, which naturally lead to new recurrence algorithms for the Class II and Class III Adomian polynomials. MATHEMATICA programs generating these classes of Adomian polynomials are subsequently presented. Computation shows that for computer generation of the Class III Adomian polynomials, the new algorithm reduces the running times compared with the definitional formula. We also consider the number of summands of these classes of Adomian polynomials and obtain the corresponding formulas. Finally, we demonstrate the versatility of the four classes of Adomian polynomials with several examples, which include the nonlinearity of the form f(t,u), explicitly depending on the argument t.
TL;DR: In this paper, the value distribution of difference polynomials of meromorphic functions was studied and the Tumura-Clunie theorems of the TUMURA-CLUDE type were extended to difference poynomials.
TL;DR: In this paper, various aspects of the Al-Salam-Chihara q-Laguerre polynomials are described, including the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients.
TL;DR: In this paper, the first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a ($k-1$)th-order analysis are established.
Abstract: A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to $k$th-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a ($k-1$)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.
TL;DR: In this paper, all quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation.
Abstract: We show how all the quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation. Furthermore, the prepotential need not be assumed a priori. The prepotential, the deforming function, the potential, the eigenfunctions and eigenvalues are all derived within the same framework. The exceptional polynomials are expressible as a bilinear combination of a deformation function and its derivative.
TL;DR: A new class of generalized Apostol–Bernoulli polynomials is introduced based on a definition given by Natalini and Bernardini (2003) and a generalization of the Srivastava–Pinter addition theorem is obtained.
TL;DR: In this article, a Riemann-Hilbert approach to the theory of matrix orthogonal poly-nomials is presented, where the authors focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonality polynomials.
Abstract: We give a Riemann{Hilbert approach to the theory of matrix orthogonal poly- nomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of lad- der operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.
TL;DR: A family of polynomials is introduced that is dual of the Charlier polynoms by the Stirling transform, and the resulting combinatorial identities for the number of partitions of a set into subsets of size at least $2$.
Abstract: We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least $2$.
TL;DR: In this article, the Bochner theorem for first order operators of Dunkl type was established for polynomial solutions, and it was shown that the only families of orthogonal polynomials in this category are limits of little and big $q$-Jacobi polynomorphisms as $q=-1$.
Abstract: We establish an analogue of the Bochner theorem for first order operators of Dunkl type, that is we classify all such operators having polynomial solutions. Under natural conditions it is seen that the only families of orthogonal polynomials in this category are limits of little and big $q$-Jacobi polynomials as $q=-1$.
TL;DR: In this article, the authors give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomial, and give some formulae useful in the rapidly developing so-called free probability.
TL;DR: By using the action of a linear functional L on a polynomial p ( x ) Sheffer sequences and Appell sequences, some fundamental properties of the Genocchi polynomials are obtained.
TL;DR: The representation of the Schrodinger group in one space dimension is explicitly constructed in the basis of the harmonic oscillator states as mentioned in this paper, and the underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues' formula, etc) that these matrix orthogonal polynomials satisfy.
Abstract: The representations of the Schrodinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states. These representations are seen to involve matrix orthogonal polynomials in a discrete variable that have Charlier and Meixner polynomials as building blocks. The underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues’ formula, etc) that these matrix orthogonal polynomials satisfy.