Open Access

On σ-type zero of Sheffer polynomials

Journal of Inequalities and Applications20132013:241

https://doi.org/10.1186/1029-242X-2013-241

Received: 29 January 2013

Accepted: 19 April 2013

Published: 14 May 2013

The Erratum to this article has been published in Journal of Inequalities and Applications 2015 2015:89

Abstract

The main object of this paper is to investigate some properties of σ-type polynomials in one and two variables.

MSC:33C65, 33E99, 44A45, 46G25.

Keywords

Appell sets differential operator Sheffer polynomials generalized Sheffer polynomials

1 Introduction

In 1945, Thorne [1] obtained an interesting characterization of Appell polynomials by means of the Stieltjes integral. Srivastava and Manocha [2] discussed the Appell sets and polynomials. Dattoli et al. [3] studied the properties of the Sheffer polynomials. Recently Pintér and Srivastava [4] gave addition theorems for the Appell polynomials and the associated classes of polynomial expansions and some cases have also been discussed by Srivastava and Choi [5] in their book.

Appell sets may be defined by the following equivalent condition: { P n ( x ) } , n = 0 , 1 , 2 , is an Appell set [68] ( P n being of degree exactly n) if either
  1. (i)

    P n ( x ) = P n 1 ( x ) , n = 0 , 1 , 2 ,  , or

     
  2. (ii)
    there exists a formal power series A ( t ) = n = 0 a n t n ( a 0 0 ) such that
    A ( t ) exp ( x t ) = n = 0 P n ( x ) t n .
     

Sheffer’s A-type classification

Let ϕ n ( x ) be a simple set of polynomials and let ϕ n ( x ) belong to the operator
J ( x , D ) = k = 0 T k ( x ) D k + 1 ,

with T k ( x ) of degree ≤k. If the maximum degree of the coefficients T k ( x ) is m, then the set ϕ n ( x ) is of Sheffer A-type m. If the degree of T k ( x ) is unbounded as k , we say that ϕ n ( x ) is of Sheffer A-type ∞.

Polynomials of Sheffer A-type zero

Let ϕ n ( x ) be of Sheffer A-type zero. Then ϕ n ( x ) belong to the operator
J ( D ) = k = 0 c k D k + 1 ,
in which c k are constants. Here c 0 0 and J ϕ n = ϕ n 1 . Furthermore, since c k are independent of x for every k, a function J ( t ) exists with the formal power series expansion
J ( t ) = k = 0 c k t k + 1 , c 0 0 .
Let H ( t ) be the formal inverse of J ( t ) ; that is,
H ( J ( t ) ) = J ( H ( t ) ) = t .

Theorem (Rainville [9])

A necessary and sufficient condition that ϕ n ( x ) be of Sheffer A-type zero is that ϕ n ( x ) possess the generating function indicated in
A ( t ) exp ( x H ( t ) ) = n = 0 ϕ n ( x ) t n ,
in which H ( t ) and A ( t ) have (formal) expansions
H ( t ) = n = 0 h n t n + 1 , h 0 0 , A ( t ) = n = 0 a n t n , a 0 0 .

Theorem (Al-Salam and Verma [10])

Let { P n ( x ) } be a polynomial set. In order for { P n ( x ) } to be a Sheffer A-type zero, it is necessary and sufficient that there exist (formal) power series
H ( t ) = j = 1 h j t j , h 1 0 , A s ( t ) = j = 0 a j ( s ) t j ( not all a 0 ( s ) are zero )
and
j = 1 r A j ( t ) exp ( x H ( ε j t ) ) = n = 0 P n ( x ) t n ,
where
J ( D ) P n ( x ) = P n r ( x ) ( n = r , r + 1 , ) where J ( D ) = k = 0 a k D k + r , a 0 0

and r is a fixed positive integer. The function A ( t ) may be called the determining function for the set { P n ( x ) } .

Polynomial of σ-type zero [9, 11]

Let { p n ( x ) } be a simple set of polynomials that belongs to the operator
J ( x , σ ) = k = 0 T k ( x ) σ k + 1 , σ = D i = 1 q ( x D + b i 1 ) , D = d d x , ( J ( x , σ ) p n ( x ) = p n 1 ( x ) ) ,

where b i are constants, not equal to zero or a negative integer, and T k ( x ) are polynomials of degree ≤k. We can say that this set is of σ-type m if the maximum degree of T k ( x ) is m, m = 0 , 1 , 2 ,  .

A necessary and sufficient condition that ϕ n ( x ) be of σ-type zero, with
σ = D i = 1 q ( x D + b i 1 ) ,
is that ϕ n ( x ) possess the generating function
A ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( t ) ) = n = 0 ϕ n ( x ) t n ,
in which H ( t ) and A ( t ) have (formal) expansions
H ( t ) = n = 0 h n t n + 1 , h 0 0 ,
and
A ( t ) = n = 0 a n t n , a 0 0 .

Since ϕ n ( x ) belongs to the operator J ( σ ) = k = 0 c k σ k + 1 , where c k are constant and c 0 0 .

2 Main results

Theorem 1 If p n ( x ) is a polynomial set, then p n ( x ) is of σ-type zero with σ = D m = 1 q ( x D + b m 1 ) . It is necessary and sufficient condition that there exist formal power series
H ( t ) = n = 0 h n t n + 1 , h 0 0 ,
and
A i ( t ) = n = 0 a n ( i ) t n ( not all a 0 ( i ) are zero )
such that
i = 1 r A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = n = 0 p n ( x ) t n ,
(1)

where θ = x D .

Proof Let y i = 0 F q ( ; b 1 , b 2 , , b q ; z i ) , where i = 1 , 2 , , r , be a solution of the following differential equation:
[ θ m = 1 q ( x D + b m 1 ) z i ] y i = 0 , θ = x D , D = d d x .
On substituting z i = x H ( ε i t ) and keeping t as a constant, where
σ = D m = 1 q ( x D + b m 1 ) , θ = x D ,
we get
[ x D m = 1 q ( θ + b m 1 ) x H ( z i ) ] y i = 0 .
This can also be written as
σ y i = H ( ε i t ) y i
or
σ 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = H ( ε i t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) .
Operating J ( σ ) on both sides of Equation (1) yields
J ( σ ) n = 0 p n ( x ) t n = J ( σ ) i = 1 r A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = i = 1 r A i ( t ) J ( H ( ε i t ) ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = t n = 0 p n ( x ) t n = n = 1 p n 1 ( x ) t n .

Therefore, J ( σ ) p 0 ( x ) = 0 and J ( σ ) p n ( x ) = p n 1 ( x ) , n 1 .

Since J ( σ ) is independent of x, using the definition of σ-type [9, 11], we arrive at the conclusion that p n ( x ) is σ-type zero.

Conversely, suppose p n ( x ) is of σ-type zero and belongs to the operator J ( σ ) . Now q n ( x ) is a simple set of polynomials, we can write
i = 1 r 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = n = 0 p n ( x ) t n ,
(2)

where ε 1 , ε 2 , , ε r are the roots of unity.

Since q n ( x ) is a simple set, there exists a sequence c k [10], independent of n, such that
p n ( x ) = k = 0 n c n k q k ( x )
and
n = 0 p n ( x ) t n = n = 0 k = 0 n c n k q k ( x ) t n .
On replacing n by n + k , this becomes
n = 0 p n ( x ) t n = n = 0 k = 0 c n q k ( x ) t n + k = k = 0 q k ( x ) t k n = 0 c n t n .
Setting c n = a n ( i ) (i is independent of n, where i = 1 , 2 , , r ), this becomes
n = 0 p n ( x ) t n = k = 0 q k ( x ) t k n = 0 a n ( i ) t n , by using Equation (2), we get = i = 1 r A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) .

This completes the proof. □

Theorem 2 A necessary and sufficient condition that p n ( x ) be of σ-type zero and there exist a sequence h k , independent of x and n, such that
i = 1 r ε i n h n 1 ψ ( ε i t ) = σ p n ( x ) ,
(3)

where ψ ( ε i t ) = A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) .

Proof If p n ( x ) is of σ-type zero, then it follows from Theorem 1 that
n = 0 p n ( x ) t n = i = 1 r A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) .
This can be written as
n = 0 σ p n ( x ) t n = i = 1 r A i ( t ) σ 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = i = 1 r H ( ε i t ) A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = i = 1 r ( n = 0 h n ε i n + 1 t n + 1 ) A i ( t ) 0 F q ( ; b 1 , b 2 , , b q ; x H ( ε i t ) ) = n = 1 i = 1 r ( ε i n h n 1 ) ψ ( ε i t ) t n .
Thus
σ p n ( x ) = i = 1 r ε i n h n 1 ψ ( ε i t ) .

This completes the proof. □

3 Sheffer polynomials in two variables [12]

Let p n ( x , y ) be of σ-type zero. Then p n ( x , y ) belongs to an operator J ( σ ) = k = 0 c k σ k + 1 , in which c k are constants and c 0 0 .

Since
J ( σ ) p n ( x , y ) = p n 1 ( x , y ) , n 1 ,
where
D x = x , D y = y , θ = x x , ϕ = y y , σ x = D x m = 1 p ( θ + b m 1 ) , σ y = D y s = 1 q ( θ + b s 1 ) ,
and
J ( ( G + H ) ( t ) ) = ( ( G + H ) J ( t ) ) = t , σ = σ x + σ y .
Theorem 3 A necessary and sufficient condition that p n ( x , y ) be of σ-type zero, with
σ x = D x m = 1 p ( θ + b m 1 ) , σ y = D y s = 1 q ( θ + b s 1 ) , σ = σ x + σ y ,
is that p n ( x , y ) possess a generating function in
i = 1 r A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = n = 0 p n ( x , y ) t n ,
(4)
in which
G ( t ) = n = 0 g n t n + 1 , g 0 0 , H ( t ) = n = 0 h n t n + 1 , h 0 0 , A i ( t ) = n = 0 a n ( i ) t n ( not all a 0 ( i ) are zero )

and i is independent of n.

Proof Let u i = 0 F p ( ; b 1 , b 2 , , b p ; z i ) and v i = 0 F q ( ; c 1 , c 2 , , c q ; w i ) be the solutions of the following differential equations:
[ θ z m = 1 p ( θ z + b m 1 ) z i ] u i = 0 , θ z = z z ,
and
[ ϕ w s = 1 q ( ϕ w + c s 1 ) z i ] w i = 0 , ϕ w = w w .
On substituting z i = x G ( ε i t ) , w i = y H ( ε i t ) and keeping t as a constant, where θ = x x = θ z , ϕ = y y = ϕ w , we get
θ m = 1 p ( θ + b m 1 ) u i = x G ( ε i t ) u i
and
ϕ s = 1 q ( ϕ + c s 1 ) w i = y H ( ε i t ) w i .
This can also be written as
σ 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = { G ( ε i t ) + H ( ε i t ) } 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) .
Operating J ( σ ) on both sides of Equation (4) yields
J ( σ ) n = 0 p n ( x , y ) t n = J ( σ ) i = 1 r A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = i = 1 r A i ( t ) J ( ( G + H ) ( ε i t ) ) 0 F p [ ; b 1 , b 2 , , b p ; x G ( ε i t ) ] 0 F q [ ; c 1 , c 2 , , c q ; y H ( ε i t ) ] = t n = 0 p n ( x , y ) t n = n = 1 p n 1 ( x , y ) t n .

Therefore, J ( σ ) p 0 ( x , y ) = 0 and J ( σ ) p n ( x , y ) = p n 1 ( x , y ) , n 1 .

Since J ( σ ) is independent of x and y, thus we arrive at the conclusion that p n ( x , y ) is of σ-type zero.

Conversely, suppose p n ( x , y ) is of σ-type zero and belongs to the operator J ( σ ) . Now q n ( x , y ) is a simple set of polynomials. We can write
i = 1 r 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = n = 0 p n ( x , y ) t n .
(5)
Since q n ( x , y ) is a simple set, there exists a sequence c k , independent of n, such that
p n ( x , y ) = k = 0 n c n k q k ( x , y )
and
n = 0 p n ( x , y ) t n = n = 0 k = 0 n c n k q k ( x , y ) t n .
On replacing n by n + k , this becomes
= n = 0 k = 0 c n q k ( x , y ) t n + k = k = 0 q k ( x , y ) t k n = 0 c n t n .
Setting c n = a n ( i ) (i is independent of n, where i = 1 , 2 , , r ), this becomes
= k = 0 q k ( x , y ) t k n = 0 a n ( i ) t n = i = 1 r A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) .

This completes the proof. □

Theorem 4 A necessary and sufficient condition that p n ( x , y ) be of σ-type zero and there exist sequences g k and h k , independent of x, y and n, such that
i = 1 r ε i n ( g n 1 + h n 1 ) υ ( ε i t ) = σ p n ( x , y ) ,
(6)

where υ ( ε i t ) = A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) .

Proof If p n ( x , y ) is of σ-type zero, then it follows from Theorem 3 that
n = 0 p n ( x , y ) t n = i = 1 r A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) .
This can be written as
n = 0 σ p n ( x , y ) t n = i = 1 r A i ( t ) σ 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = i = 1 r ( G + H ) ( ε i t ) A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) × 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = i = 1 r ( n = 0 ( g n + h n ) ε i n + 1 t n + 1 ) A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) × 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) = n = 1 i = 1 r ( ε i n ( g n 1 + h n 1 ) ) υ ( ε i t ) t n .
Thus
σ p n ( x , y ) = i = 1 r ε i n ( g n 1 + h n 1 ) υ ( ε i t ) ,

where υ ( ε i t ) = A i ( t ) 0 F p ( ; b 1 , b 2 , , b p ; x G ( ε i t ) ) 0 F q ( ; c 1 , c 2 , , c q ; y H ( ε i t ) ) . This completes the proof. □

Notes

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are indebted to the referees for their valuable suggestions which led to a better presentation of paper.

Authors’ Affiliations

(1)
Department of Mathematics, S. V. National Institute of Technology

References

  1. Thorne CJ: A property of Appell sets. Am. Math. Mon. 1945, 52: 191–193. 10.2307/2305676MathSciNetView ArticleMATHGoogle Scholar
  2. Srivastava HM, Manocha HL: A Treatise on Generating Functions. Wiley, New York; 1984.MATHGoogle Scholar
  3. Dattoli G, Migliorati M, Srivastava HM: Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Math. Comput. Model. 2007, 45(9–10):1033–1041. 10.1016/j.mcm.2006.08.010MathSciNetView ArticleMATHGoogle Scholar
  4. Pintér Á, Srivastava HM: Addition theorems for the Appell polynomials and the associated classes of polynomial expansions. Aequ. Math. 2012. doi:10.1007/s00010–012–0148–8Google Scholar
  5. Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.MATHGoogle Scholar
  6. Galiffa, JD: The Sheffer B-type 1 orthogonal polynomial sequences. Ph.D. Dissertation, University of Central Florida (2009)Google Scholar
  7. Goldberg JL:On the Sheffer A-type of polynomials generated by A ( t ) ψ ( x B ( t ) ) . Proc. Am. Math. Soc. 1966, 17: 170–173.MathSciNetMATHGoogle Scholar
  8. Sheffer IM: Note on Appell polynomials. Bull. Am. Math. Soc. 1945, 51: 739–744. 10.1090/S0002-9904-1945-08437-7MathSciNetView ArticleMATHGoogle Scholar
  9. Rainville ED: Special Functions. Macmillan Co., New York; 1960.MATHGoogle Scholar
  10. Al Salam WA, Verma A: Generalized Sheffer polynomials. Duke Math. J. 1970, 37: 361–365. 10.1215/S0012-7094-70-03746-4MathSciNetView ArticleMATHGoogle Scholar
  11. Mc Bride EB: Obtaining Generating Functions. Springer, New York; 1971.View ArticleGoogle Scholar
  12. Alidad, B: On some problems of special functions and structural matrix analysis. Ph.D. Dissertation, Aligarh Muslim University (2008)Google Scholar

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© Shukla et al.; licensee Springer. 2013

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