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  • Erratum
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Erratum: On σ-type zero of Sheffer polynomials

Journal of Inequalities and Applications20152015:89

https://doi.org/10.1186/s13660-015-0605-8

  • Received: 3 February 2015
  • Accepted: 3 February 2015
  • Published:

The original article was published in Journal of Inequalities and Applications 2013 2013:241

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
$$B_{n}(x+y)= \sum_{k=0}^{n} {n \choose k} B_{n-k}(x) B_{k}(y), \quad n=0,1,2,\ldots, $$
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
$$P_{n}(x+y)= \sum_{k=0}^{n} {n \choose k} B_{n-k}(x) P_{k}(y),\quad n=0,1,2,\ldots. $$
The correct theorem is given as follows.

Theorem 2

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
$$ \sum_{k=0}^{n-1} \sum _{i = 1}^{r} \bigl( \varepsilon_{i}^{k+1}h_{k} \bigr) p_{n-k-1}(x)= \sigma p_{n}(x), $$
(3)
where \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{r}\) are roots of unity and r is a fixed positive integer.

Proof

If \(p_{n}(x)\) is of σ-type zero, then it follows from Theorem 1 (see [1]) that
$$\sum_{n = 0}^{\infty}p_{n}(x)t^{n} = \sum_{i = 1}^{r}A_{i}(t) {}_{0}F_{q}\bigl(-;b_{1},b_{2}, \ldots,b_{q};xH(\varepsilon_{i}t)\bigr). $$
This can be written as
$$\begin{aligned} \sum_{n = 0}^{\infty} \sigma p_{n}(x)t^{n} &= \sum_{i = 1}^{r}A_{i}(t) \sigma {}_{0}F_{q}\bigl(-;b_{1},b_{2}, \ldots,b_{q};xH(\varepsilon_{i}t)\bigr) \\ &=\sum_{n=0}^{\infty} \sum _{k=0}^{n} \sum_{i = 1}^{r} \bigl( \varepsilon _{i}^{k+1}h_{k} \bigr) p_{n-k}(x)t^{n+1} =\sum_{n=1}^{\infty} \sum _{k=0}^{n-1} \sum_{i = 1}^{r} \bigl( \varepsilon _{i}^{k+1}h_{k} \bigr) p_{n-k-1}(x)t^{n}. \end{aligned}$$
Thus
$$\sigma p_{n}(x)= \sum_{k=0}^{n-1} \sum_{i = 1}^{r} \bigl( \varepsilon _{i}^{k+1}h_{k} \bigr) p_{n-k-1}(x). $$
This gives the proof of the statement. □

The correct theorem is given as follows.

Theorem 4

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
$$ \sigma p_{n}(x,y)= \sum_{k=0}^{n-1} \sum_{i = 1}^{r} \varepsilon_{i}^{k+1} ( g_{k} + h_{k} ) p_{n-k-1}(x,y), $$
(6)
where \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{r}\) are roots of unity and r is a fixed positive integer.

Proof

If \(p_{n}(x,y)\) is of σ-type zero, then it follows from Theorem 3 (see [1]) that
$$\sum_{n = 0}^{\infty}p_{n}(x,y)t^{n} = \sum_{i = 1}^{r}A_{i}(t) {}_{0}F_{p}\bigl(-;b_{1},b_{2}, \ldots,b_{p};xG(\varepsilon_{i}t)\bigr) {}_{0}F_{q} \bigl(-;c_{1},c_{2},\ldots,c_{q};yH( \varepsilon_{i}t)\bigr). $$
This can be written as
$$\begin{aligned} \sum_{n = 0}^{\infty} \sigma p_{n}(x,y)t^{n} &= \sum_{i = 1}^{r}A_{i}(t) \sigma{}_{0}F_{p}\bigl(-;b_{1},b_{2}, \ldots,b_{p};xG(\varepsilon_{i}t)\bigr) {}_{0}F_{q} \bigl(-;c_{1},c_{2},\ldots,c_{q};yH( \varepsilon_{i}t)\bigr) \\ &=\sum_{n=0}^{\infty} \sum _{k=0}^{n} \sum_{i = 1}^{r} \varepsilon _{i}^{k+1} ( g_{k} + h_{k} ) p_{n-k}(x,y)t^{n+1} \\ &=\sum_{n=1}^{\infty} \sum _{k=0}^{n-1} \sum_{i = 1}^{r} \varepsilon _{i}^{k+1} ( g_{k} + h_{k} ) p_{n-k-1}(x,y)t^{n}. \end{aligned}$$
Thus
$$\sigma p_{n}(x,y)= \sum_{k=0}^{n-1} \sum_{i = 1}^{r} \varepsilon_{i}^{k+1} ( g_{k} + h_{k} ) p_{n-k-1}(x,y). $$
This is the proof of Theorem 4. □

Notes

Declarations

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.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Department of Mathematics, S.V. National Institute of Technology, Surat, 395 007, India

References

  1. 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 View ArticleMathSciNetGoogle Scholar
  2. Brown, JW: On multivariable Sheffer sequences. J. Math. Anal. Appl. 69, 398-410 (1979) View ArticleMATHMathSciNetGoogle Scholar

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