Search

A simple closed form for a particular instance of Wilson’s loop variables is derived both via group theory and via q-hypergeometric series. Several facets of these diverse approches are explored.

This paper is a survey of recent work on orthogonal polynomials and Chevalley groups. The orthogonal polynomials we emphasize are those given by basic hypergeometric series, or q-series. The group theoretic significance of these polynomials is that they are the spherical functions for Chevalley groups over the finite field GF(q).

One looks for [formal] orthogonal polynomials satisfying interesting differential or difference relations and equations (Laguerre-Hahn theory). The divided difference operator used here is essentially the Askey-Wilson operator $$Df\left( x \right) = \frac{{E_2 f\left( x \right) - E_1 f\left( x \right)}}{{E_2 x - E_1 x}} = \frac{{f\left( {y_2 \left( x \right)} \right) - f\left( {y_1 \left( x \right)} \right)}}{{y_2 \left( x \right) - y_1 \left( x \right)}}$$ where y1(x) and y2(x) are the two roots of Ay2+2Bxy+Cx2++2Dy+2Ex+f=0.

We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. Here these ideas are applied to the (continuous) Askey-Wilson polynomials, introduced by Askey and Wilson, to obtain a strikingly simple derivation of their orthogonality relations. 1. In t roduct ion . In [1] the authors and A.K. Agarwal introduced symmetry techniques for the study of families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. The fundamental objects in this study are the recurrence relations obeyed by the families: generating functions and identities for each family are characterized in terms of the recurrence relations. The functions studied in [1] are of the form ^ / a i , . . . , a r \ _ y > (ai;q)k'"(ar;q)k z k V 6 i , . . . , 6 8 ' ' / f^Q(bi;q)k'"(b8;q)k{q;q)k and many-variable extension. Here , . _ J 1 if k = 0 ( f l ; g , ' " \ ( l a ) ( l g a ) . . . ( l c * 1 o ) if fc = 1,2,... and \q\ . ( , , . ) ( . ( I ^ ) ( I ^ " E.G. KALNINS AND W. MILLER, JR. 225 Here Eff(z) = f(qz). Relation (2.1A) follows from T(az;q)J-;q) = J ^ ( l q)(aqz;q)k.1 (^— ;q) \Z /k q' \ Z / k-l and (2.IB) follows from KaZ;q)k(±q)k = (1 a b q ^ ^ q ) ^ ^ q ) ^ The first relation was pointed out by Askey and Wilson [2]; we have not found (2. IB) in the literature. Consider relation (2.IB); it suggests that there should be an operator Aaq-^,bq-^,cq,dq) . Aa,b,c,d) T , , a A l± mapping Cn t o

We consider a partial-wave expansion for $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\omega}$ in terms of the spin-matrix polynomials ${W}_{n}(z)=\frac{\ensuremath{\Gamma}(z+\frac{1}{2}n+\frac{1}{2})}{\ensuremath{\Gamma}(z\ensuremath{-}\frac{1}{2}n+\frac{1}{2})}$ rather than the conventional Legendre polynomials. This leads naturally to Veneziano's formula when his supplementary condition $\ensuremath{\alpha}(s)+\ensuremath{\alpha}(t)+\ensuremath{\alpha}(u)=2$ is invoked.

n/a

Some further results on possible root locations of two-dimensional (2-D) polynomials are presented. It is established that the distinguished boundary of the unit bidisc being free of zeros is a nongeneric property of 2-D polynomials. The result is illustrated by a simple example.

(1909). On the Irreducibility of Certain Polynomials. The American Mathematical Monthly: Vol. 16, No. 4, pp. 66-67.

Let S be a set of difference polynomials. The perfect difference ideal {S} [2, p. 76 and p. 82] may properly contain the difference ideal ~/[S]. It follows that in determining the irreducible components of the manifold of S it is not sufficient to consider only factorizations of polynomials of n/IS ] (or, equivalently, of IS]). In particular, let S contain only the algebraically irreducible difference polynomial F, and let M be a singular component [2, p. 161] of the manifold ofF. M will be called strongly singular if it can be found from factorizations in [F], and weakly singular otherwise. This definition will be made precise in Section 1. By analogy, all singular components of a differential polynomial with coefficients in a field of characteristic 0 should be considered strongly singular, since if G is such a polynomial, ~/[G] = {G}. An examination of the previously known examples of singular components of difference polynomials reveals that they are all strongly singular. This is not surprising in itself, since it appears to be difficult to prove that a manifold is a component if it is weakly singular. However, the examples cover three rather large classes of singular components, namely, those described in Theorems I, II, and III of Section 1, as well as one example with markedly different properties [2, p. 332]. It would indicate a most unexpected simplicity in the structure of the perfect ideals generated by algebraically irreducible difference polynomials if the known examples were an indication of the general situation. The principal object of this paper is to remove this possibility by providing an example (Section 2) of a difference polynomial F with the weakly singular component M = {0}. This example is of order 5, and Theorem I shows that it could not be of order less than 3. Whether examples of orders 3 or 4 exist is not known. In Section 3 approximations to solutions in singular components are considered for the case of coefficients and solutions which are complex valued functions on the integers. Using the polynomial Q of the example of Section 2 it is shown that the solution 0 constituting the singular component can be-approximated uniformly by solutions which are nowhere 0. This is quite unexpected. (Compare [3, p. 259].) It is easily seen to be possible only because of two circumstances: 1) {0} is a weakly singular component, 2) the approximating solutions are not regular, that is they do not lie in difference rings which are integral domains. Although no general theorem can be stated at present, this example suggests an important distinction in the behavior of weakly and strongly singular components, The distinction between weak and strong singularity is only one of an infinite sequence of similar distinctions that may be made using, in place of the difference ideal [F] generated by the difference polynomial F, each ideal of the sequence whose union is {F}, as described in [2, Ch. 3, Section 2].

A five-parameter family of reproducing kernels is constructed for q-Racah polynomials. Special cases for q-Hahn and little q-Jacobi polynomials are considered by selecting the parameters appropriately. Corresponding bilinear sums are also obtained for a whole range of q-orthogonal polynomials. As a special case, some product formulas are obtained for q-Racah and q-Wilson polynomials.

The following q-integral Xd(-qt1c)c,-j(qt1d),o-j d t -c (-qet)ca+j dq is evaluated. A more general q-integral is also considered. Some applications to the q-Wilson (or Askey-Wilson) polynomials are also given.

1. In a joint paper with A. C. Schaeffer1 we discussed the following question: Let D be a closed domaina in the complex z-plane and z0 a fixed pointt of D. Let its consider all polynomtials f(z) of givez degree n forwhich f Jf(z) ? 1 in D and f(z0) is real. What is the largest possible value of !f (z) if f(z) is an arbitrary polynomial of the hinzd mentioned and z is arbitrary in D? Under certain conditions on D and zo the inequality I sf(z)! 0 clepends only oi 1D and zo. This theorem was well-known in the special case when D is a circle anld zO is ani in-terior point of D. fHowever Mr. P. Erd6s has called my attention to the fact that the best possible bound of j 3:tf(z) j is nlot known in this case. In the present paper I wish to determine the best possible bounid in case D is a circle ancd zo is the center of D. I prove the following

In this study, we investigate the polynomials for and positive integers k and a positive real number a, with the initial values , We give some fundamental properties related to them. Also, we obtain asymptotic results for the roots of polynomials . MSC:11B39, 11B37.

The main purpose of this paper is by using the generating function methods and some combinatorial techniques to establish some new recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials. It turns out that some known results in (He and Wang in Adv. Differ. Equ. 2012:209, 2012) are obtained as special cases. MSC: 11B68; 05A19

A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P, the algebra of polynomials in a single variable x over all linear functional on P. By using the way of the umbral algebra, we obtain some fundamental properties of the generalized Apostol-type polynomials. We also show some special cases which include the corresponding results of Dere and Simsek etc. MSC:05A40, 11B68, 05A10, 05A15.