Shukla et al. Journal of Inequalities and Applications (2015) 2015:89 DOI 10.1186/s13660-015-0605-8

d Journal of Inequalities and Applications

a SpringerOpen Journal

ERRATUM

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Erratum: On a -type zero of Sheffer polynomials

Ajay K Shukla*, Shrinivas J Rapeli and PratikVShah

"Correspondence: ajayshukla2@rediffmail.com Department of Mathematics, S.V. Nationalinstitute of Technology, Surat, 395 007, India

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 {Bn(x)} is a polynomial sequence which is simple and of degree precisely n. {Bn(x)} is a binomial sequence if

Bn(x + y) = ¿ (nW(x)Bk(y), n = 0, 1, 2, ...,

k=0 ^ '

and a simple polynomial sequence {Pn(x)} is a Sheffer sequence if there is a binomial sequence {Bn(x)} such that

Pn(x + y) = £ (nWn(x)Pn(y), n = 0, 1, 2, ....

k=0 ^ '

The correct theorem is given as follows.

Theorem 2 A necessary and sufficient condition that pn(x) be of a-type zero and there exists a sequence hk independent ofx and n such that

mm(ek+1hn)Pn-n-i(x) = apn(x),

n=0 1=1

where e1, e2,...,er are roots of unity and r is a fixed positive integer. Proof Ifpn(x) is of a-type zero, then it follows from Theorem 1 (see [1]) that

Y^Pn(x)f = Ai(t)0Fq(--, bi, b2,...,bq; xH M).

n=0 i=1

This can be written as

ft Spri

^apn(x)tn = ^ Ai(t)a 0Fq(-; bi, b2,...,bq; xH (sit))

œ n r

œ n-1 r

= EE E(sf+1 hk)Pn-n (x)tn+1 = ££ J2(s'n+1h^Pn-n-i(x)tn.

n=0 n=0 i=1

n=1 n=0 i=1

ringer

© 2015 Shukla et al.; licensee Springer. 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.

Shukla et al. Journal of Inequalities and Applications (2015) 2015:89

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a Pn (x) = ^Yh hk)pn-k-l(x).

k=0 i=l

This gives the proof of the statement. □

The correct theorem is given as follows.

Theorem 4 A necessary and sufficient condition thatpn (x, y) be symmetric, a class ofpoly-nomials in two variables and a-type zero, there exists a sequence gk and hk, independent ofx, y and n such that

a pn (x, y) = ^ ^£k+l(gk + hk )pn-k-i(x, y), (6)

k=0 i=l

where sl, s2,...,sr are roots of unity and r is a fixed positive integer.

Proof If pn (x, y) is of a-type zero, then it follows from Theorem 3 (see [l]) that

!> (x, y)tn = ^2 Ai (t)o Fp(-; bl, b2,..., bp; xG(^t))o Fq(-; ci, c2,..., cq; yH fet)).

n=0 i=l

This can be written as

^ a pn (x, y)tn = ^ Ai (t)a o Fp(-; bl, b2,..., bp; xG(^t))o F^-; ci, c2,..., cq; yH (stf))

n=0 i=l

TO n r

=E E Egk+l(g*+hk )pn-k (x, y)tn+l

n=0 k=0 i=l TO n-l r

=E E Eek+l(gk+hk )pn-k-l(x, y)tn.

n=l k=0 i=l

a pn (x, y) = ^ +l(gk + hk )pn-k-l(x, y).

k=0 i=l

This is the proof of Theorem 4. □

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.

Received: 3 February 2015 Accepted: 3 February 2015 Published online: 06 March 2015 References

1. Shukla, AK, Rapeli, SJ, Shah, PV: On a -type zero of Sheffer polynomials. J. Inequal. Appl. 2013,241 (2013). doi:10.1186/1029-242X-2013-241

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