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On σ-type zero of Sheffer polynomials
Journal of Inequalities and Applications volume 2013, Article number: 241 (2013)
Abstract
The main object of this paper is to investigate some properties of σ-type polynomials in one and two variables.
MSC:33C65, 33E99, 44A45, 46G25.
1 Introduction
In 1945, Thorne [1] obtained an interesting characterization of Appell polynomials by means of the Stieltjes integral. Srivastava and Manocha [2] discussed the Appell sets and polynomials. Dattoli et al. [3] studied the properties of the Sheffer polynomials. Recently Pintér and Srivastava [4] gave addition theorems for the Appell polynomials and the associated classes of polynomial expansions and some cases have also been discussed by Srivastava and Choi [5] in their book.
Appell sets may be defined by the following equivalent condition: , is an Appell set [6–8] ( being of degree exactly n) if either
-
(i)
, , or
-
(ii)
there exists a formal power series () such that
Sheffer’s A-type classification
Let be a simple set of polynomials and let belong to the operator
with of degree ≤k. If the maximum degree of the coefficients is m, then the set is of Sheffer A-type m. If the degree of is unbounded as , we say that is of Sheffer A-type ∞.
Polynomials of Sheffer A-type zero
Let be of Sheffer A-type zero. Then belong to the operator
in which are constants. Here and . Furthermore, since are independent of x for every k, a function exists with the formal power series expansion
Let be the formal inverse of ; that is,
Theorem (Rainville [9])
A necessary and sufficient condition that be of Sheffer A-type zero is that possess the generating function indicated in
in which and have (formal) expansions
Theorem (Al-Salam and Verma [10])
Let be a polynomial set. In order for to be a Sheffer A-type zero, it is necessary and sufficient that there exist (formal) power series
and
where
and r is a fixed positive integer. The function may be called the determining function for the set .
Polynomial of σ-type zero [9, 11]
Let be a simple set of polynomials that belongs to the operator
where are constants, not equal to zero or a negative integer, and are polynomials of degree ≤k. We can say that this set is of σ-type m if the maximum degree of is m, .
A necessary and sufficient condition that be of σ-type zero, with
is that possess the generating function
in which and have (formal) expansions
and
Since belongs to the operator , where are constant and .
2 Main results
Theorem 1 If is a polynomial set, then is of σ-type zero with . It is necessary and sufficient condition that there exist formal power series
and
such that
where .
Proof Let , where , be a solution of the following differential equation:
On substituting and keeping t as a constant, where
we get
This can also be written as
or
Operating on both sides of Equation (1) yields
Therefore, and , .
Since is independent of x, using the definition of σ-type [9, 11], we arrive at the conclusion that is σ-type zero.
Conversely, suppose is of σ-type zero and belongs to the operator . Now is a simple set of polynomials, we can write
where are the roots of unity.
Since is a simple set, there exists a sequence [10], independent of n, such that
and
On replacing n by , this becomes
Setting (i is independent of n, where ), this becomes
This completes the proof. □
Theorem 2 A necessary and sufficient condition that be of σ-type zero and there exist a sequence , independent of x and n, such that
where .
Proof If is of σ-type zero, then it follows from Theorem 1 that
This can be written as
Thus
This completes the proof. □
3 Sheffer polynomials in two variables [12]
Let be of σ-type zero. Then belongs to an operator , in which are constants and .
Since
where
and
Theorem 3 A necessary and sufficient condition that be of σ-type zero, with
is that possess a generating function in
in which
and i is independent of n.
Proof Let and be the solutions of the following differential equations:
and
On substituting , and keeping t as a constant, where , , we get
and
This can also be written as
Operating on both sides of Equation (4) yields
Therefore, and , .
Since is independent of x and y, thus we arrive at the conclusion that is of σ-type zero.
Conversely, suppose is of σ-type zero and belongs to the operator . Now is a simple set of polynomials. We can write
Since is a simple set, there exists a sequence , independent of n, such that
and
On replacing n by , this becomes
Setting (i is independent of n, where ), this becomes
This completes the proof. □
Theorem 4 A necessary and sufficient condition that be of σ-type zero and there exist sequences and , independent of x, y and n, such that
where .
Proof If is of σ-type zero, then it follows from Theorem 3 that
This can be written as
Thus
where . This completes the proof. □
References
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors are indebted to the referees for their valuable suggestions which led to a better presentation of paper.
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The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.
An erratum to this article is available at http://dx.doi.org/10.1186/s13660-015-0605-8.
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Shukla, A.K., Rapeli, S.J. & Shah, P.V. On σ-type zero of Sheffer polynomials. J Inequal Appl 2013, 241 (2013). https://doi.org/10.1186/1029-242X-2013-241
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DOI: https://doi.org/10.1186/1029-242X-2013-241