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Erratum: On σ-type zero of Sheffer polynomials

  • 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.

FormalPara 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.

FormalPara 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.

FormalPara 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.

FormalPara 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. □

References

  1. 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

    Article  MathSciNet  Google Scholar 

  2. 2.

    Brown, JW: On multivariable Sheffer sequences. J. Math. Anal. Appl. 69, 398-410 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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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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Correspondence to Ajay K Shukla.

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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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