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Orthogonality of the Meixner-Pollaczek polynomials beyond Favard's theorem.

1 Introduction

First, let us fix some terminologies, notations and conventions that we will use throughout this paper.

The set of complex numbers will be denoted by C and i will stand for the imaginary unit ([i.sup.2] = -1); the set of positive integers will be denoted by N, and N0 will denote the set of nonnegative integers. All polynomials considered will be complex-valued in one complex variable, and P will stand for the set of all such polynomials. For each n [member of] [N.sub.0], the subset of P of all polynomials of degree not greater than n will be denoted by [P.sub.n]. By a system of monic polynomials we will mean a sequence [{[P.sub.n]}.sup.[infinity].sub.n=0] of polynomials satisfying [P.sup.(n).sub.n] = n! for each n [member of] [N.sub.0]. For notational convenience, we will use [P.sub.-1] to denote the null polynomial.

For n [member of] N, a (square) matrix of order n, with complex entries [a.sub.jk], will be denoted by A = [([a.sub.jk]).sup.n-1.sub.j,k=0] (the entry [a.sub.jk] will be also called the (j + 1, k + 1)th element of the matrix A), and [([a.sub.j]).sup.n-1.sub.j=0] [member of] [C.sup.n] will stand for the matrix of order 1 x n (equivalently, for the vector) ([a.sub.0], [a.sub.1], ..., [a.sub.n-1]). The conjugate transpose of a matrix A = [([a.sub.jk]).sup.n-1.sub.j,k=0] will be denoted by using the superscript * (with or without parenthesis, as needed), that is,

[A.sup.*] = [(A).sup.*] = [([a.sub.jk]).sup.n-1*.sub.j,k=0] = [([([a.sub.jk]).sup.n-1.sub.j,k=0]).sup.*] = [([[bar.a].sub.kj]).sup.n-1.sub.j,k=0],

(the overline denotes, of course, complex conjugation). A square matrix A will be called Hermitian whenever A = [A.sup.*]; a Hermitian matrix A will be called positive definite whenever [xAx.sup.*] > 0 for each x [member of] [C.sup.n]\{0} (as usual, we will identify the only (element o)f (a matrix) of order 1 with the matrix itself, so the matrix ([([x.sub.j]).sup.n-1.sub.j=0])([([a.sub.jk]).sup.n-1.sub.j,k=0])[([([x.sub.j]).sup.n-1.sub.j=0]).sup.*] will be identified with its unique entry [[summation].sup.n-1.sub.j=0][[summation].sup.n-1.sub.k=0][a.sub.jk][x.sub.j][bar.[x.sub.k]] Hermitian matrices, positive definiteness is equivalent to the requirement that all of its principal minors are positive, and also equivalent to the fact that all its eigenvalues are positive. A sesqui/inearform in a linear complex space V is a map (*, *) : V x V [right arrow] C that is linear in its first (left) argument and conjugate-linear in the second (right) one; when this sesquilinear form is positive definite (i.e., when (x, x) > 0 for each x [member of] [C.sup.n]\{0}) the map is called an inner product in V.

The Kronecker delta will be denoted by [[delta].sub.ij], and [(*).sub.n] will denote the so-called shifted factorial (also, Pochhammer symbol), defined by

[(x).sub.0] = 1, [(x).sub.n+1] = x(x + 1) ... (x + n), n [member of] [N.sub.0], x [member of] C.

As usual, the binomial coefficient for complex numbers n, k is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the hypergeometric series [sub.m][F.sub.n] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [([b.sub.1]).sub.k], ..., [([b.sub.n]).sub.k] [not equal to] 0 for all k [member of] [N.sub.0]. When m = 0 (n = 0) the numerator (denominator) of [([a.sub.1]).sub.k] ... [([a.sub.m]).sub.k]/[([b.sub.1]).sub.k] ... [([b.sub.n]).sub.k] becomes 1. Clearly, if one of the numerator parameters satisfies -[a.sub.j] [member of] [N.sub.0], then the hypergeometric series is a polynomial of degree min{-[a.sub.j]: -[a.sub.j] [member of] [N.sub.0]}.

In concluding this first part of the introduction, we recall that the nth iteration of an operator [PSI] : P [right arrow] P is recursively defined by means of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Now, let us make a brief survey of non-standard orthogonality in the literature.

By a non-standard orthogona/ity result we will mean an orthogonality statement for a system of monic polynomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with parameters [[lambda].sub.1], ..., [[lambda].sub.m], and satisfying the three term recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), for those values of the parameters for which [b.sub.n] vanishes for some n [greater than or equal to] 1. This topic has attracted great interest in recent years. In [11], Kwon and Littlejohn state that for each N [member of] N, the La guerre polynomials [{[L.sup.(-N).sub.n]}.sup.[infinity].sub.n=0] form an orthogonal sequence with respect to a positive-definite inner product that can be written as a discrete-continuous bilinear form involving derivatives. A unified approach to the orthogonality of the (generalized) Laguerre polynomials [{[L.sup.([alpha]).sub.n]}.sup.[infinity].sub.n=0], for arbitrary real a, can be found in [20]. For a given positive integer number N, the orthogonality of the generalized Gegenbauer polynomials [{[C.sup.(-N+1/2).sub.n]}.sup.[infinity].sub.n=0] is solved in [3], using again a Sobolev inner product, that is, an inner product involving derivatives (the case N = 1 is considered also in [12]). Other cases for Jacobi polynomials are solved in [2], where the families [{[[P.sub.n].sup.(-N,[beta]).sub.n]}.sup.[infinity].sub.n=0], N [member of] N, -(N + [beta]) [not member of] N and [{[P.sup.([alpha],-N).sub.n]}.sup.[infinity].sub.n=0] , N [member of] N, - ([alpha] + N) [not member of] N are considered, and also in [1], in which the orthogonality for [{[P.sup.(-N,-M).sub.n]}.sup.[infinity].sub.n=0], N, M [member of] N is stated. The orthogonality of the sequence [{[M.sup.([gamma],[mu]).sub.n]}.sup.[infinity].sub.n=0] of generalized Meixner polynomials, with [gamma] [member of] R and 0 < [mu] < 1, is given in [4], where a special consideration is taken for the values [gamma] = 1 - N, N [member of] N. A non-standard inner product with respect to which the symmetric Meixner-Pollaczek polynomials [{[P.sup.([lambda]).sub.n](*/2;n[pi]/2)}.sup.[infinity].sub.n=0] ([lambda] [member of] R) become orthogonal is introduced in [6]. For (not necessarily symmetric) generalized Meixner Pollaczek polynomials [{[P.sup.(0).sub.n](*; [phi])}.sup.[infinity].sub.n=0] (0 < [phi] < [pi]) we can find an orthogonality result in [8].

In this paper, we consider suitable modifications of our previous result [14, Theorem 3], adapted to the case of Meixner-Pollaczek polynomials [{[P.sup.([lambda]).sub.n](*; [phi])}.sup.[infinity].sub.n=0] with arbitrary complex parameter [lambda], in order to state the orthogonality of the families [{[P.sup.((1-N)/2).sub.n](*; [phi])}.sup.[infinity].sub.n=0], where N [member of] N, the parameters (1 - N)/2 being the only ones for which no orthogonality condition is ensured by Favard's theorem. For analogous results in the q-world we refer the reader to [15, 16, 17, 18, 19].

The paper is organized as follows. In Section 2 we extend the monic Meixner Pollaczek polynomial system [{[P.sup.([lambda]).sub.n](*; [phi])}.sup.[infinity].sub.n=0], classically defined for [lambda] > 0 and 0 < [phi] < [pi], giving an explicit definition that works perfectly well for all [lambda] [member of] C; we will also give in this section two preparatory results, one concerning the roots of the new polynomials, and the other one concerning the action of the iterations of the linear operator [delta]/[delta]x on them. In Section 3 we define a nonstandard discrete-continuous inner product which yields orthogonality for the extended monic Meixner-Pollaczek polynomials with those "exceptional" values of the parameter [lambda] for which Favard's theorem fails to work, i.e. for [lambda] [member of] {0, -1/2, -1, -3/2, ...}.

2 The generalized Meixner-Pollaczek polynomials

J. Meixner [13] introduced in 1934 a class of polynomials that F. Pollaczek [21] considered independently sixteen years later. This remarkable family of orthogonal polynomials is a generalization of some of the classical ones, and it exhibits in many aspects a singular behaviour (for a brief but enlightening discussion, see [22, pp. 393-400]). These so-called Meixner-Pollaczek polynomials appear in the Askey-scheme of hypergeometric orthogonal polynomials [7, 9].

For each [lambda] > 0 and each [phi] [member of] (0, [pi]), the nth degree monic Meixner-Pollaczek polynomial [P.sup.([lambda]).sub.n](*; [phi]) can be defined in terms of the hypergeometric series [sub.2][F.sub.1] by means of (see (1.7.1) and (1.7.4) in [9])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)

and they satisfy the three term recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

where

[a.sup.([lambda],[phi]).sub.n] = -n + [lambda]/tan [phi], [b.sup.([lambda],[phi]).sub.n] = n(n + 2[lambda] - 1)/4 [sin.sup.2][phi], (2.3)

with the agreement that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The orthogonality condition is (see (1.7.2) and (1.7.4) in [9])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.4)

where m, n [member of] [N.sub.0], [lambda] > 0 and 0 < [phi] < [pi].

Our intention is to accomplish the extension of the monic Meixner-Pollaczek polynomials [{[P.sup.([lambda]).sub.n](*; [phi])}.sub.[infinity].sub.n=0] for all complex values of the parameter [lambda]. We first observe that (2.1) does not hold when -2[lambda] [member of] [N.sub.0], but after straightforward manipulation of the series we obtain an expression for [{[P.sup.([lambda]).sub.n](*; [phi])}.sup.[infinity].sub.n=0] which is defined for all [lambda] [member of] C, [phi] [member of] (0, [pi]) and n [member of] [N.sub.0].

Definition 2.1. For each [lambda] [member of] C, each [phi] [member of] (0, [pi]) and for all n [member of] [N.sub.0] we define the nth degree monic generalized Meixner-Pollaczek polynomials [P.sup.([lambda]).sub.n](*; [phi]) by means of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

Observe that this new extended family satisfies the same three term recurrence relation (2.2), (2.3) as the classical monic Meixner-Pollaczek polynomials with positive parameter [lambda]. Hence, taking into account that for n [greater than or equal to] 1 one has [b.sup.([lambda],[phi]).sub.n] = 0 only when -2[lambda] [member of] [N.sub.0], the corresponding orthogonality statement would be that the generalized family of monic Meixner-Pollaczek polynomials is orthogonal with respect to a quasi-definite moment functional (which is positive definite if [lambda] > 0) if and only if -2[lambda] [member of] C\N0.

Now we will give some results that will be essential in the main results of this paper.

As shown in [5, Proposition 6], [P.sup.(0).sub.n] (x/2); [pi]/2) = (x/2)[P.sup.(1).sub.n-1](x/2; [pi]/2) for n [greater than or equal to] 1, where we have adapted the original relation to our normalization. In [8, Proposition 13], the author improves this relation and gets (again, in the version of monic polynomials) [P.sup.(0).sub.n](x; [phi]) = x[P.sup.(1).sub.n-1](x; [phi]), n [greater than or equal to] 1. The following result generalizes these ones.

Proposition 2.1. Let N [member of] N. For each integer n > N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.6)

Proof. For the sake of brevity we introduce the notation

[C.sup.(n;[phi]).sub.k] = [e.sup.in[phi]][(1 - [e.sup.-2i[phi]]).sup.k]/[(2 sin [phi]).sup.n], n [member of] [N.sub.0], 0 [less than or equal to] k [less than or equal to] n, 0 < [phi] < [pi].

Using (2.5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [(-N + 1 + k).sub.n-k] = (-N + k + 1)(-N + k + 2) ... (-N + n), if n [greater than or equal to] N (which implies that the last factor in the shifted factorial is a non-negative integer), then for each k [less than or equal to] N - 1 (which implies the non-positiveness of the first factor) we have [(-N + 1 + k).sub.n-k] = 0. Consequently, for n [greater than or equal to] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking into account that for 0 [less than or equal to] k [less than or equal to] n - N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Noting that

[P.sup.((1-N)/2).sub.N](x; [phi]) = [(-1).sup.N][C.sup.(N;[phi]).sub.N][(1 - N/2 + ix).sub.N] = [(-i).sup.N][(1 - N/2 + ix).sub.N'] (2.7)

we finally establish the factorization (2.6).

Corollary 2.1. For fixed N [member of] N, let us denote [x.sup.(N).sub.k] = (((2k + 1) - N)/2)i for 0 [less than or equal to] k [less than or equal to] N - 1. We have

[P.sup.((1-N)/2).sub.n]([x.sup.(N).sub.k]; [phi]) = 0, 0 [less than or equal to] k [less than or equal to] N - 1, n [greater than or equal to] N, 0 < [phi] < [pi].

Proof. Since for 0 [less than or equal to] k [less than or equal to] N - 1 the points [x.sup.(N).sub.k] are the N different (pure imaginary) roots of the equation [((1 - N)/2 + ix).sub.N] = 0, from (2.6) we deduce that for n [greater than or equal to] N, each [x.sup.(N).sub.k] is a root of the polynomial [P.sup.((1-N)/2).sub.n](*; [phi]).

We define the forward shift operator [delta]: P [right arrow] P as usual (see [9, 0.9.1]), that is, for each polynomial p, the polynomial [delta](p) := [delta]p is the one defined by means of [delta]p(x) = p(x + i/2) - p(x - i/2), x [member of] C. It is clear that 5 is a linear operator that reduces by one the degree of the evaluated polynomial. For the power functions [e.sub.n], defined by [e.sub.n](x) = [x.sup.n] for n [member of] [N.sub.0] and x [member of] C, it is usual to denote [delta][e.sub.n](x) = [delta][x.sup.n]. Then [delta]x = i, and also ([delta]p(x))/([delta]x) = (1/i)[delta]p(x). Thus, the symbol [delta]/[delta]x will stand for the linear operator -i[delta].

Using [9, 1.7.7] (in its version for monic polynomials) in (2.5), we can easily verify that the same forward shift relation holds for the generalized monic Meixner-Pollaczek polynomials. That is to say

[delta][P.sup.([lambda]).sub.n](x; [phi])/[delta]x = n[P.sup.([lambda]+1/2).sub.n-1](x; [phi]), n [member of] [N.sub.0], [lambda], x [member of] C.

(where we have used [delta][P.sup.([lambda]).sub.n](x; [phi]) instead of the more formal [delta][P.sup.([lambda]).sub.n] (*; [phi])(x), and we will use this convention in the sequel). Therefore,

Proposition 2.2. Given a fixed nonnegative integer k, we have, for each n [greater than or equal to] k - 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Orthogonality of [{[P.sup.((1-N)/2).sub.n](*; [phi])}.sup.[infinity].sub.n=0] for positive integers N

By using the Poisson kernel and the associated Poisson measure, T.K. Araaya [5] gives an orthogonality result for the system [{[P.sup.([lambda]).sub.n](*/2; [pi]/2)}.sup.[infinity].sub.n=0] of symmetric Meixner-Pollaczek polynomials with parameter [lambda] = 0. This result was further extended by the same author [6] to arbitrary real values of the parameter, by introducing a non-standard inner product with respect to which the symmetric Meixner-Pollaczek polynomials become orthogonal. In both cases, the technique of the proofs depends strongly both on the use of the generating function (to replace the sequence of symmetric Meixner-Pollaczek polynomials) and on some nice calculus machinery. As far as we know, D. Dominici [8, Remark 17] gives the first non-standard orthogonality statement for (not necessarily symmetric) generalized Meixner-Pollaczek polynomials with parameter [lambda] = 0.

With the aid of a suitable modification of our result [14, Theorem 3], adapted to the case considered here of generalized monic Meixner-Pollaczek polynomials, we can give orthogonality results that are intimately related with those of Araaya and Dominici.

Definition 3.1. Let N [member of] N and let [phi] [member of] (0, [pi]). For a Hermitian and positive definite complex matrix A of order N, and for the points [x.sub.k] = [x.sup.(N).sub.k] = (((2k + 1) -- N)/2)i, 0 [less than or equal to] k [less than or equal to] N - 1, we define the inner product [(*, *).sub.(N;A;[phi])] : P x P [right arrow] R by means of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.8)

Our aim is to show that there exist matrices A such that the sequence of generalized monic Meixner-Pollaczek polynomials with parameter [lambda] equal to (1 - N)/2 is orthogonal with respect to the non-standard inner product introduced above.

Theorem 3.1. For a fixed positive integer N and for [phi] [member of] (0, [pi]), the generalized monic Meixner-Pollaczek polynomials [{[P.sup.((1-N)/2).sub.n](*; [phi])}.sup.[infinity].sub.n=0] are orthogonal with respect to the inner product [(*, *).sub.(N;A;[phi]),] where A = [C.sup.-1]D[([C.sup.-1]).sup.*], C stands for the matrix [([P.sup.((1-N)/2).sub.n]([x.sub.k]; [phi])).sup.N-1.sub.j,k=0] and D is an arbitrary diagonal matrix of order N with positive entries in its diagonal.

Proof. First we must verify that A is Hermitian and positive definite.

Let [{[l.sub.j]}.sup.N-1.sub.j=0] [subset] [P.sub.N-1] be the set of Lagrange interpolating polynomials with respect to the points [{[x.sub.k]}.sup.N-1.sub.k=0] (that is, [l.sub.j]([x.sub.k]) = [[delta].sub.jk] for 0 [less than or equal to] j, k [less than or equal to] N - 1}). Taking into account that both [{[P.sup.((1-N)/2).sub.j](*; [phi])}.sup.N-1.sub.j=0] and [{[l.sub.j]}.sup.N-1.sub.j=0] are bases of [P.sub.N-1], and also that

[P.sup.((1-N)/2).sub.j](x;[phi]) = [N-1.summation over (k=0)][P.sup.((1-N)/2).sub.j]([x.sub.k]; [phi])[l.sub.k](x), 0 [less than or equal to] j [less than or equal to] N - 1,

we can justify that the matrix C = [([P.sup.((1-N)/2).sub.j]([x.sub.k]; [phi])).sup.N-1.sub.j,k=0] is nonsingular.

We recall that a square complex matrix M is positive definite if and only if there exists a nonsingular complex matrix Q such that M = [QQ.sup.*]. Thus, A = [C.sup.-1]D[([C.sup.-1]).sup.*] = ([C.sup.-1][square root of D])[([C.sup.-1][square root of D]).sup.*] is positive definite (here, stands for the diagonal matrix whose diagonal entries are the positive square root of the corresponding diagonal entries of D).

Now we will state the orthogonality in three steps:

i) In case that 0 [less than or equal to] m, n [less than or equal to] N - 1, since

[([delta]/[delta]x).sup.N][P.sup.((1-N)/2).sub.m](x; [phi]) = [([delta]/[delta]x).sup.N][P.sup.((1-N)/2).sub.n](x; [phi]) = 0,

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[kappa].sub.n] is the (positive) (n + 1, n + 1)th entry of the matrix D.

ii) If 0 [less than or equal to] m [less than or equal to] N - 1 and n [greater than or equal to] N, then

[([delta]/[delta]x).sup.N][P.sup.((1-N)/2).sub.m](x; [phi]) = 0, and [P.sup.((1-N)/2).sub.n]([x.sub.k]; [phi]) = 0, 0 [less than or equal to] k [greater than or equal to] N - 1,

so, clearly, we have ([P.sup.((1-N)/2).sub.m](*; [phi]),[P.sup.((1-N)/2).sub.n][(*; [phi])).sub.(N;A;[phi])] = 0.

iii) Finally, when m, n [greater than or equal to] N, using that [P.sup.(1/2).sub.n-N](x; [phi]) is real for x [member of] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us note that if we choose D = [([j!.sup.2][(2 sin [phi]).sup.2(N-j)-1][[delta].sub.jk]).sup.N-1.sub.j,k=0] in the previous theorem, then we have the closed form for the norms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We now give a new related orthogonality result in which the discrete part of the sesquilinear form (3.8) changes in such a way that symmetry is gained and explicitness is lost.

Theorem 3.2. Fixed N [member of] N and [phi] [member of] (0, [pi]), there exists a Hermitian and positive definite matrix A of order N such that the family [{[P.sup.((1-N)/2).sub.n](*; [phi])}.sup.[infinity].sub.n=0] is orthogonal with respect to the inner product [(*, *).sub.([delta];N;A;[phi])] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.9)

Proof.

A simple induction argument shows that for each k [member of] [N.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, by Corollary 2.1 we have, fixed a nonnegative integer N and for 0 [less than or equal to] k [less than or equal to] N - 1 and n [greater than or equal to] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now consider the set of fundamental polynomials [{[h.sub.j]}.sup.N-1.sub.j=0] [subset] [P.sub.N-1] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly, these polynomials exist, are unique, and form a basis of [P.sub.N-1]. Taking into account that both [{[P.sup.((1-N)/2).sub.j](*; [phi])}.sup.N-1.sub.j=0] and [{[h.sub.j]}.sup.N-1.sub.j=0] are bases of [P.sub.N-1], there exists a nonsingular matrix [C.sup.t] = [([c.sub.jk]).sup.N-1.sub.j,k=0] such that for each j = 0, 1, ..., N - 1,

[P.sup.((1-N)/2).sub.j](x; [phi]) = [N-1.summation over (k=0)][c.sub.kj][h.sub.k](x),

which implies that C = [([[delta].sup.k][P.sup.((1-N)/2).sub.j]([x.sub.k] - (k/2)i; [phi])).sup.N-1.sub.j,k=0].

Then, for any arbitrary diagonal matrix D of order N with positive elements in the diagonal, and defining A = [C.sup.-1]D[([C.sup.-1]).sup.*], we get the desired conclusion following the same reasoning as in Theorem 3.2, replacing in steps i) and ii) [P.sup.((1-N)/2).sub.m]([x.sub.k]; [phi]) and [P.sup.((1-N)/2).sub.n]([x.sub.k]; [phi]) by [[delta].sup.k][P.sup.((1- N)/2).sub.m]([x.sub.k] - (k/2)i; [phi]) and [[delta].sup.k][P.sup.((1-N)/2).sub.n]([x.sub.k] - (k/2)i; [phi]), respectively.

As a final remark, let us say that defining [??]: P [right arrow] P by [??]p(x) = p(x + i) - p(x) we get a result similar to the previous one, where [[delta].sup.k][P.sup.((1-N)/2).sub.n] ([x.sub.k] - (k/2)i; [phi]) is replaced by [[??].sup.k][P.sup.((1-N)/2).sub.n](((1 - N)/2)i; [phi]) and where C = [([[??].sup.k][P.sup.((1-N)/2).sub.j](((1 - N)/2)i; [phi])).sup.N-1.sub.j,k=0].

Acknowledgements

We thank the referees for their useful suggestions and comments, which have improved the presentation of the paper, and for bringing reference [10] to our attention

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Departamento de Matematicas, Universidad de Jaen, 23071 Jaen, Spain.

e-mail: samuel@ujaen.es, emgarcia@ujaen.es

* This work was partially supported by Junta de Andalucla, Research Group FQM 0178, and by Ministerio de Educacion y Ciencia, Project MTM2006-14590.

Received by the editors January 2012--In revised form in April 2012.

Communicated by A. Bultheel.

2000 Mathematics Subject Classification: 33C45, 42C05.
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Author:Moreno, Samuel G.; Garcia-Caballero, Esther M.
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Jan 1, 2013
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