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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*.

If \(p_{n}(x)\) 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*.

If \(p_{n}(x,y)\) 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-2013-241Brown, 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