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Erratum: On σ-type zero of Sheffer polynomials
Journal of Inequalities and Applications volume 2015, Article number: 89 (2015)
After publication of our work [1], we realized that there are some mathematical errors in Theorem 2 and Theorem 4. Our aim is to correct and modify Theorems 2 and 4.
Brown [2] discussed that \(\lbrace B_{n}(x) \rbrace\) is a polynomial sequence which is simple and of degree precisely n. \(\lbrace B_{n}(x) \rbrace\) is a binomial sequence if
and a simple polynomial sequence \(\lbrace P_{n}(x) \rbrace\) is a Sheffer sequence if there is a binomial sequence \(\lbrace B_{n}(x) \rbrace\) such that
The correct theorem is given as follows.
A necessary and sufficient condition that \(p_{n}(x)\) be of σ-type zero and there exists a sequence \(h_{k} \) independent of x and n such that
where \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{r}\) are roots of unity and r is a fixed positive integer.
FormalPara ProofIf \(p_{n}(x)\) is of σ-type zero, then it follows from Theorem 1 (see [1]) that
This can be written as
Thus
This gives the proof of the statement. □
The correct theorem is given as follows.
A necessary and sufficient condition that \(p_{n}(x,y)\) be symmetric, a class of polynomials in two variables and σ-type zero, there exists a sequence \(g_{k} \) and \(h_{k} \), independent of x, y and n such that
where \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{r}\) are roots of unity and r is a fixed positive integer.
FormalPara ProofIf \(p_{n}(x,y)\) is of σ-type zero, then it follows from Theorem 3 (see [1]) that
This can be written as
Thus
This is the proof of Theorem 4. □
References
Shukla, AK, Rapeli, SJ, Shah, PV: On σ-type zero of Sheffer polynomials. J. Inequal. Appl. 2013, 241 (2013). doi:10.1186/1029-242X-2013-241
Brown, JW: On multivariable Sheffer sequences. J. Math. Anal. Appl. 69, 398-410 (1979)
Acknowledgements
Authors are grateful to Prof. MEH Ismail for his comments and suggestions. The second author is thankful to SVNIT, Surat, India for awarding JRF and SRF.
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The online version of the original article can be found under doi:10.1186/1029-242X-2013-241.
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Shukla, A.K., Rapeli, S.J. & Shah, P.V. Erratum: On σ-type zero of Sheffer polynomials. J Inequal Appl 2015, 89 (2015). https://doi.org/10.1186/s13660-015-0605-8
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DOI: https://doi.org/10.1186/s13660-015-0605-8