Book Title
Abstract
In this contribution, we study the orthogonality conditions satisfied by AlSalamCarlitz polynomials when the parameters and are not necessarily real nor ‘classical’, i.e., the linear functional with respect to such polynomial sequence is quasidefinite and not positive definite. We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the AlSalamCarlitz polynomials of degree up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.
Chapter 1 The orthogonality of AlSalamCarlitz polynomials for complex parameters
H. S. Cohl, R. S. CostasSantos and W. Xu] Howard S. Cohl, Roberto S. CostasSantos and Wenqing Xu
Applied and Computational Mathematics Division,
National Institute of Standards and Technology,
Gaithersburg, MD 208998910, USA
howard.cohl@nist.gov
Departamento de Física y Matemáticas, Facultad de Ciencias, Universidad de
Alcalá, 28871 Alcalá de Henares, Madrid, Spain
rscosa@gmail.com
Department of Mathematics and Statistics, California
Institute of Technology, CA 91125, USA
williamxuxu@yahoo.com
Keywords: orthogonal polynomials; difference operator; integral
representation; discrete measure.
MSC classification: 33C45; 42C05
1 Introduction
The AlSalamCarlitz polynomials were introduced by W. A. AlSalam and L. Carlitz in [1] as follows:
(0) 
In fact, these polynomials have a Rodriguestype formula [2, (3.24.10)]
where
the Pochhammer symbol (shifted factorial) is defined as
and the derivative operator is defined by
Remark 1
Observe that by the definition of the derivative
The expression (1) shows us that is an analytic function for any complex value parameters and , and thus can be considered for general .
The classical AlSalamCarlitz polynomials correspond to parameters and . For these parameters, the AlSalamCarlitz polynomials are orthogonal on with respect to the weight function . More specifically, for and [2, (14.24.2)],
where
and the Jackson integral [2, (1.15.7)] is defined as
where
Taking into account the previous orthogonality relation, it is a direct result that if and are classical, i.e., , , with , all the zeros of are simple and belong to the interval , but this is no longer valid for general and complex. In this paper we show that for general , complex numbers, but excluding some special cases, the AlSalamCarlitz polynomials may still be characterized by orthogonality relations. The case and or and are classical, i.e., the linear functional with respect to such polynomial sequence is orthogonal is positive definite and in such a case there exists a weight function so that
Note that this is the key for the study of many properties of AlSalamCarlitz polynomials I and II. Thus, our goal is to establish orthogonality conditions for most of the remaining cases for which the linear form is quasidefinite, i.e., for all
We believe that these new orthogonality conditions can be useful in the study of the zeros of AlSalamCarlitz polynomials. For general , the zeros are not confined to a real interval, but they distribute themselves in the complex plane as we can see in Figure 1. Throughout this paper denote .
2 Orthogonality in the complex plane
Theorem 1
Let , , , the AlSalamCarlitz polynomials are the unique polynomials (up to a multiplicative constant) satisfying the property of orthogonality
(0) 
Remark 2
I if the lattice is a set of points which are located inside on a single contour that goes from 1 to 0, and then from 0 to through the spirals
where , which we can see in Figure 2. Taking into account ( ‣ 1), we need to avoid the case. For the case, we cannot apply Favard’s result [3], because in such a case this polynomial sequence fulfills the recurrence relation [2]
Let , and , . We are going to express the Jackson integral ( ‣ 1) as the difference of the two infinite sums and apply the identity
(0) 
Let . Then, for one side since , and using the identities [2, (14.24.7), (14.24.9)], one has
Following an analogous process as before, and since , we have
Therefore, if , and since is finite one can first repeat the previous process times obtaining
and
Hence since the difference of both limits, term by term, goes to 0 since , then
For , following the same idea, we have
since it is known that in this case [2, (14.24.2)]
Due to the normality of this polynomial sequence, i.e., for all , the uniqueness is straightforward, hence the result holds. From this result, and taking into account that the squared norm for the AlSalamCarlitz polynomials is known, we got the following consequence for which we could not find any reference.
Corollary 1
Let , . Then
The following case, which is just the AlSalamCarlitz polynomials for the case, is commonly called the AlSalamCarlitz II polynomials.
Theorem 2
Let , , . Then, the AlSalamCarlitz polynomials are unique (up to a multiplicative constant) satisfying the property of orthogonality given by
(0) 
Let us denote by , then . For , . Then, by using the identity (2) replacing , and taking into account that and [2, (14.24.9)], for one has
Following the same idea from the previous result, we have
Therefore, the property of orthogonality holds for . Next, if , we have
Using the same argument as in Theorem 1, the uniqueness holds, so the claim follows.
Remark 3
Observe that in the previous theorems if , with , , after some logical cancellations, the set of points where we need to calculate the integral is easy to compute. For example, if and , one obtains the sum [2, p. 537, (14.25.2)].
Remark 4
The case is special because it is not considered in the literature. In fact, the linear form associated with the AlSalamCarlitz polynomials is quasidefinite and fulfills the Pearsontype distributional equations
Moreover, the AlSalamCarlitz polynomials fulfill the threeterm recurrence relation [2, (14.24.3)]
(0) 
where with initial conditions , .
Therefore, we believe that it will be interesting to study such a case for its peculiarity because the coefficient for all , so one can apply Favard’s result.
2.1 The case.
In this section we only consider the case where is a root of unity. Let be a positive integer such that then, due to the recurrence relation ( ‣ 4) and following the same idea that the authors did in [4, Section 4.2], we apply the following process:

The sequence is orthogonal with respect to the Gaussian quadrature
where are the zeros of for such value of .

Since , we need to modify such a linear form.
Next, we can prove that the sequence is orthogonal with respect to the bilinear form
since .

Since and taking into account what we did before, we consider the linear form

Therefore one can obtain a sequence of bilinear forms such that the AlSalamCarlitz polynomials are orthogonal with respect to them.
3 A generalized generating function for AlSalamCarlitz polynomials
For this section, we are going to assume , or . Indeed, by starting with the generating functions for AlSalamCarlitz polynomials [2, (14.25.1112)], we derive generalizations using the connection relation for these polynomials.
Theorem 3
Let , , . Then
(0) 
If we consider the generating function for AlSalamCarlitz polynomials [2, (14.25.11)]
and multiply both sides by , obtaining
(0) 
If we now apply the binomial theorem [2, (1.11.1)]
to ( ‣ 3), and then collect powers of , we obtain
Taking into account this expression, the result follows.
Theorem 4
Let , , , , . Then
(0) 
where
is the unilateral basic hypergeometric series.
We start with a generating function for AlSalamCarlitz polynomials [2, (14.25.12)]
and ( ‣ 3) to obtain
If we reverse the order of summations, shift the variable by a factor of , using the basic properties of the Pochhammer symbol, and [2, (1.10.1)]. Observe that we can reverse the order of summation since our sum is of the form
where
In this case, one has
and , where , , and are positive constants independent of . Therefore, if , then
and this completes the proof. As we saw in Section 2, the orthogonality relation for AlSalamCarlitz polynomials for , , and is
Taking this result in mind, the following result follows.
Theorem 5
Let , , , , . Then
From ( ‣ 4), we replace and multiply both sides by , and by using the orthogonality relation (2), the desired result holds.
Note that the application of connection relations to the rest of the known generating functions for AlSalamCarlitz polynomials [2, (14.24.11), (14.25.11)] leave these generating functions invariant.
Acknowledgments
The author R. S. CostasSantos acknowledges financial support by National Institute of Standards and Technology. The authors thank the anonymous referee for her/his valuable comments and suggestions. They contributed to improve the presentation of the manuscript.
References
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 2. R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their analogues. Springer Monographs in Mathematics, SpringerVerlag, Berlin (2010). With a foreword by Tom H. Koornwinder.
 3. T. S. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach Science Publishers, New YorkLondonParis (1978). Mathematics and its Applications, Vol. 13.
 4. R. S. CostasSantos and J. F. SánchezLara, Orthogonality of polynomials for nonstandard parameters, Journal of Approximation Theory. 163(9), 1246–1268 (2011).