• Erratum
• Open Access

Erratum: On σ-type zero of Sheffer polynomials

Journal of Inequalities and Applications20152015:89

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

• Accepted: 3 February 2015
• Published:

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

After publication of our work , 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  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 ) 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 ) 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. □

Authors’ Affiliations

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

References 