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1.
We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic beta integral for the root system of type A. Validity of this integral is established by a different method as well. 相似文献
2.
The aim of the present paper is to introduce and study a new type of q-Mellin transform [11], that will be called q-finite Mellin transform. In particular, we prove for this new transform an inversion formula and q-convolution product. The application of this transform is also earlier proposed in solving procedure for a new equation with a new fractional differential operator of a variational type. 相似文献
3.
In this article, we introduce the two dimensional Mellin transform M_(f)(s, t),give some properties, establish the Paley-Wiener theorem and Plancherel formula, present the Hausdorff-Young inequality, and find several applications for the two dimensional Mellin transform. 相似文献
4.
《Quaestiones Mathematicae》2013,36(1):23-33
Abstract Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities. 相似文献
5.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy the following two conditions:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
6.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
7.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
8.
LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence n of measures such that limnn and limn
n
A
= the limit measure is of the formf+[(1-f)]A for some (unique) Borel function 0f1Cb(A). Furthermore, conditions are given such that any such functionf occurs. 相似文献
9.
Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0?i?d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide.In this paper we show that the following (i)-(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A∗.(iv) The pair A,A∗ is determined up to isomorphism by the data , where θi (resp.) is the eigenvalue of A (resp.A∗) on Vi (resp.), and is the split sequence of A,A∗ corresponding to and . 相似文献
10.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
11.
Raimundas Vidūnas 《Linear algebra and its applications》2007,422(1):39-57
Let V denote a vector space with finite positive dimension, and let (A, A∗) denote a Leonard pair on V. As is known, the linear transformations A, A∗ satisfy the Askey-Wilson relations
12.
Wenchang Chu 《Constructive Approximation》2002,18(4):579-597
By iterating recursively the q-Saalschütz summation formula, we introduce the Saalschütz chain reactions. A general series transform, which expresses a
nonterminating bilateral series in terms of a finite multiple unilateral sum, will be established. As applications we derive,
by means of Bailey’s 6ψ6 -series identity, several bilateral transformations including one due to Milne [12]. These transformations
further yield a number of closed formulas of very well-poised bilateral basic hypergeometric series; which are closely related
to the identities obtained by Minton [13], Karlsson [11], Gasper [8], and Chu [5], [6], [7] through the partial fraction method
and divided differences. 相似文献
13.
Raimundas Vidūnas 《Discrete Mathematics》2008,308(4):479-495
It is known that if (A,A*) is a Leonard pair, then the linear transformations A, A* satisfy the Askey-Wilson relations
14.
We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation. 相似文献
15.
16.
Cristian Lenart 《Advances in Mathematics》2009,220(1):324-275
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms. 相似文献
17.
The very special algebraic properties of give rise to an extremely rich series of results in the form of polylogarithmic ladders up to the 8th order, with a probable extension to the 9th All but three of the results forn < 5 come from Kummer's equations Atn = 5 there are seven results of which only three come from Kummer s equations, the eleven results forn > 5 are all found numerically A family of single-variable functional equations up to the 5th order is developed to explain the presence of the remaining formulas forn 5, but is incapable of extension to the 6th orderThe connection between ladders and functional equations is discussed Some extremely large integers, into the trilhons are generated by the computation process, and the implications of this on the conjectured functional equations at the higher orders is considered At the present time, no equations forn 6 exist from which the numerically-determined results could be deduced analytically, and these formulas exist solely by computation using double or multiple precision accuracy 相似文献
18.
A brief review of the polylogarithmic ladder and its cyclotomic equation is followed by an application to the roots of two irreducible quintic equations. The first is unique in its class and possesses four accessible and two inaccessible valid ladders. The other equation gives rise to a single ladder determinable, at the present time, only by numerical computation, and is a member of a completely new category. 相似文献
19.
Atsushi Narukawa 《Advances in Mathematics》2004,189(2):247-267
We show the modular properties of the multiple “elliptic” gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper, we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products.We also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions. 相似文献
20.
H. T. Koelink 《Acta Appl Math》1996,44(3):295-352
Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A
66 (1990), 146–149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.Supported by a NATO-Science Fellowship of the Netherlands Organization for Scientific Research (NWO). 相似文献