Higher-order Bernoulli, Euler and Hermite polynomials

Abstract

In (Kim and Kim in J. Inequal. Appl. 2013:111, 2013; Kim and Kim in Integral Transforms Spec. Funct., 2013, doi:10.1080/10652469.2012.754756), we have investigated some properties of higher-order Bernoulli and Euler polynomial bases in $P n ={p(x)∈Q[x]|degp(x)≤n}$. In this paper, we derive some interesting identities of higher-order Bernoulli and Euler polynomials arising from the properties of those bases for $P n$.

1 Introduction

For $r∈R$, let us define the Bernoulli polynomials of order r as follows:

$( t e t − 1 ) r e x t = ∑ n = 0 ∞ B n ( r ) (x) t n n ! (see [1–18]).$
(1)

In the special case, $x=0$, $B n ( r ) (0)= B n ( r )$ are called the n th Bernoulli numbers of order r. As is well known, the Euler polynomials of order r are defined by the generating function to be

$( 2 e t + 1 ) r e x t = ∑ n = 0 ∞ E n ( r ) (x) t n n ! (see [1–10]).$
(2)

For $λ(≠1)∈C$, the Frobenius-Euler polynomials of order r are also given by

$( 1 − λ e t − λ ) r e x t = ∑ n = 0 ∞ H n ( r ) (x|λ) t n n ! (see [1, 7]).$
(3)

The Hermite polynomials are defined by the generating function to be:

$e 2 x t − t 2 = ∑ n = 0 ∞ H n (x) t n n ! (see [8–10, 19]).$
(4)

Thus, by (4), we get

$H n (x)= ( H + 2 x ) n = ∑ l = 0 n ( n l ) H n − l 2 l x l (see ),$
(5)

where $H n = H n (0)$ are called the n th Hermite numbers. Let $P n ={p(x)∈Q[x]|degp(x)≤n}$. Then $P n$ is an $(n+1)$-dimensional vector space over . In [8, 10], it is called that ${ E 0 ( r ) (x), E 1 ( r ) (x),…, E n ( r ) (x)}$ and ${ B 0 ( r ) (x), B 1 ( r ) (x),…, B n ( r ) (x)}$ are bases for $P n$. Let Ω denote the space of real-valued differential functions on $(∞,−∞)=R$. We define four linear operators on Ω as follows:

$I(f)(x)= ∫ x x + 1 f(x)dx,Δ(f)(x)=f(x+1)−f(x),$
(6)
$Δ ˜ (f)(x)=f(x+1)+f(x),D(f)(x)= f ′ (x).$
(7)

Thus, by (6) and (7), we get

$I n (f)(x)= ∑ k = 0 n ( n k ) ( − 1 ) n − l f n (x+l)(see [8, 10, 12, 13]),$
(8)

where $f 1 ′ =f, f 2 ′ = f 1 ,…, f n ′ = f n − 1$, $n∈N$.

In this paper, we derive some new interesting identities of higher-order Bernoulli, Euler and Hermite polynomials arising from the properties of bases of higher-order Bernoulli and Euler polynomials for $P n$.

2 Some identities of higher-order Bernoulli and Euler polynomials

First, we introduce the following theorems, which are important in deriving our results in this paper.

Theorem 1 

For $r∈ Z + =N∪{0}$, let $p(x)∈ P n$. Then we have

$p(x)= 1 2 r ∑ k = 0 n ∑ j = 0 r 1 k ! ( r j ) D k p(j) E k ( r ) (x).$

Theorem 2 

For $r∈ Z +$, let $p(x)∈ P n$:

1. (a)

If $r>n$, then we have

$p(x)= ∑ k = 0 n ∑ j = 0 k 1 k ! ( − 1 ) k − j ( k j ) ( I r − k p ( j ) ) B k ( r ) (x).$
2. (b)

If $r≤n$, then

$p ( x ) = ∑ k = 0 r − 1 ∑ j = 0 k 1 k ! ( − 1 ) k − j ( k j ) ( I r − k p ( j ) ) B k ( r ) ( x ) + ∑ k = r n ∑ j = 0 r 1 k ! ( − 1 ) r − j ( k j ) ( D k − r p ( j ) ) B k ( r ) ( x ) .$

Let us take $p(x)= H n (x)∈ P n$.

Then, by (5), we get

$p ( k ) ( x ) = D k p ( x ) = 2 k n ( n − 1 ) ⋯ ( n − k + 1 ) H n − k ( x ) = 2 k n ! ( n − k ) ! H n − k ( x ) .$
(9)

From Theorem 1 and (9), we can derive the following equation (10):

$H n ( x ) = 1 2 r ∑ k = 0 n { ∑ j = 0 r 1 k ! ( r j ) 2 k n ! ( n − k ) ! H n − k ( j ) } E k ( r ) ( x ) = 1 2 r ∑ k = 0 n ( n k ) 2 k [ ∑ j = 0 r ( r j ) H n − k ( j ) ] E k ( r ) ( x ) .$
(10)

Therefore, by (10), we obtain the following theorem.

Theorem 3 For $n,r∈ Z +$, we have

$H n (x)= 1 2 r ∑ k = 0 n ( n k ) 2 k [ ∑ j = 0 r ( r j ) H n − k ( j ) ] E k ( r ) (x).$

We recall an explicit expression for Hermite polynomials as follows:

$H n (x)= ∑ l = 0 [ n 2 ] ( − 1 ) l n ! l ! ( n − 2 l ) ! ( 2 x ) n − 2 l .$
(11)

By (11), we get

$H n − k (j)= ∑ l = 0 [ n − k 2 ] ( − 1 ) l ( n − k ) ! l ! ( n − k − 2 l ) ! ( 2 j ) n − k − 2 l .$
(12)

Thus, by Theorem 3 and (12), we obtain the following corollary.

Corollary 4 For $n,r∈ Z +$, we have

$H n (x)= 1 2 r ∑ k = 0 n { ∑ j = 0 r ∑ l = 0 [ n − k 2 ] ( n k ) ( r j ) 2 k ( − 1 ) l ( n − k ) ! ( 2 j ) n − k − 2 l l ! ( n − k − 2 l ) ! } E k ( r ) (x).$

Now, we consider the identities of Hermite polynomials arising from the property of the basis of higher-order Bernoulli polynomials in $P n$.

For $r>k$, by (6) and (8), we get

$I r − k H n ( x ) = ∑ l = 0 r − k ( r − k l ) ( − 1 ) r − k − l H n + r − k ( x + l ) 2 r − k ( n + 1 ) ⋯ ( n + r − k ) = ∑ l = 0 r − k ( r − k l ) ( − 1 ) r − k − l n ! H n + r − k ( x + l ) 2 r − k ( n + r − k ) ! .$
(13)

Therefore, by Theorem 2 and (13), we obtain the following theorem.

Theorem 5 For $n,r∈ Z +$, with $r>n$, we have

$H n (x)=n! ∑ k = 0 n { ∑ j = 0 k ∑ l = 0 r − k ( r − k l ) ( k j ) ( − 1 ) r − j − l H n + r − k ( j + l ) 2 r − k k ! ( n + r − k ) ! } B k ( r ) (x).$

Let us assume that $r,k∈ Z +$, with $r≤n$. Then, by (b) of Theorem 2, we get

$H n ( x ) = n ! ∑ k = 0 r − 1 { ∑ j = 0 k ∑ l = 0 r − k ( r − k l ) ( k j ) ( − 1 ) r − j − l H n + r − k ( j + l ) 2 r − k k ! ( n + r − k ) ! } B k ( r ) ( x ) + n ! ∑ k = r n { ∑ j = 0 r ( − 1 ) r − j ( r j ) 2 k − r H n + r − k ( j ) k ! ( n + r − k ) ! } B k ( r ) ( x ) .$
(14)

Therefore, by (14), we obtain the following theorem.

Theorem 6 For $n,r∈ Z +$, with $r≤n$, we have

$H n ( x ) = n ! ∑ k = 0 r − 1 { ∑ j = 0 k ∑ l = 0 r − k ( r − k l ) ( k j ) ( − 1 ) r − j − l H n + r − k ( j + l ) 2 r − k k ! ( n + r − k ) ! } B k ( r ) ( x ) + n ! ∑ k = r n { ∑ j = 0 r ( − 1 ) r − j ( r j ) 2 k − r k ! ( n + r − k ) ! H n + r − k ( j ) } B k ( r ) ( x ) .$

Remark From (12), we note that

$H n + r − k (j+l)= ∑ m = 0 [ n + r − k 2 ] ( − 1 ) m ( n + r − k ) ! m ! ( n + r − k − 2 m ) ! ( 2 j + 2 l ) n + r − k − 2 m$
(15)

and

$H n + r − k (j)= ∑ m = 0 [ n + r − k 2 ] ( − 1 ) m ( n + r − k ) ! m ! ( n + r − k − 2 m ) ! ( 2 j ) n + r − k − 2 m .$
(16)

Theorem 7 

For $n,r∈ Z +$, with $r>n$ and $p(x)∈ P n$, we have

$p(x)= ∑ k = 0 n { ∑ j = 0 k ∑ l = 0 n ( r − k ) ! S 2 ( l + r − k , r − k ) ( l + r − k ) ! k ! ( − 1 ) k − j ( k j ) p ( l ) ( j ) } B k ( r ) (x),$

where $S 2 (l,n)$ is the Stirling number of the second kind and $p ( l ) (j)= D l p(j)$.

Theorem 8 

For $n,r∈ Z +$, with $r≤n$ and $p(x)∈ P n$, we have

$p ( x ) = ∑ k = 0 r − 1 { ∑ j = 0 k ∑ l = 0 n ( r − k ) ! S 2 ( l + r − k , r − k ) ( l + r − k ) ! k ! ( − 1 ) k − j ( k j ) p ( l ) ( j ) } B k ( r ) ( x ) + ∑ k = r n { ∑ j = 0 r ( − 1 ) r − j k ! ( r j ) p ( k − r ) ( j ) } B k ( r ) ( x ) .$

Let us take $p(x)= H n (x)∈ P n$. Then, by Theorem 7 and Theorem 8, we obtain the following corollary.

Corollary 9 For $n,r∈ Z +$:

1. (a)

For $r>n$, we have

$H n (x)=n! ∑ k = 0 n { ∑ j = 0 k ∑ l = 0 n ( r − k ) ! S 2 ( l + r − k , r − k ) ( l + r − k ) ! k ! ( n − l ) ! ( − 1 ) k − j ( k j ) 2 l H n − l ( j ) } B k ( r ) (x).$
2. (b)

For $r≤n$, we have

$H n ( x ) = n ! ∑ k = 0 r − 1 { ∑ j = 0 k ∑ l = 0 n ( r − k ) ! S 2 ( l + r − k , r − k ) ( l + r − k ) ! k ! ( n − l ) ! ( − 1 ) k − j ( k j ) 2 l H n − l ( j ) } B k ( r ) ( x ) + n ! ∑ k = r n { ∑ j = 0 r ( − 1 ) r − j ( r j ) 2 k − r H n − k + r ( j ) k ! ( n − k + r ) ! } B k ( r ) ( x ) .$

Theorem 10 

For $p(x)∈ P n$, we have

$p(x)= 1 ( 1 − λ ) r ∑ k = 0 n { ∑ j = 0 r 1 k ! ( r j ) ( − λ ) r − j p ( k ) ( j ) } H k ( r ) (x|λ).$

Let us take $p(x)= H n (x)∈ P n$. Then

$H n ( x ) = 1 ( 1 − λ ) r ∑ k = 0 n { ∑ j = 0 r 1 k ! ( r j ) ( − λ ) r − j 2 k n ! ( n − k ) ! H n − k ( j ) } H k ( r ) ( x | λ ) = 1 ( 1 − λ ) r ∑ k = 0 n ( n k ) 2 k { ∑ j = 0 r ( r j ) ( − λ ) r − j H n − k ( j ) } H k ( r ) ( x | λ ) = 1 ( 1 − λ ) r ∑ k = 0 n { ∑ j = 0 r ∑ l = 0 [ n − k 2 ] ( n k ) 2 k ( r j ) ( − λ ) r − j ( − 1 ) l ( n − k ) ! ( 2 j ) n − k − 2 l l ! ( n − k − 2 l ) ! } H k ( r ) ( x | λ ) .$
(17)

Therefore, by (17), we obtain the following corollary.

Corollary 11 For $n∈ Z +$, we have

$H n (x)= 1 ( 1 − λ ) r ∑ k = 0 n { ∑ j = 0 r ∑ l = 0 [ n − k 2 ] ( n k ) 2 k ( r j ) ( − λ ) r − j ( − 1 ) l ( n − k ) ! ( 2 j ) n − k − 2 l l ! ( n − k − 2 l ) ! } H k ( r ) (x|λ).$

For $r=1$, the Frobenius-Euler polynomials are defined by the generating function to be

$( 1 − λ e t − λ ) e x t = ∑ n = 0 ∞ H n (x|λ) t n n ! (see ).$
(18)

Thus, by (18), we get

$d d λ H n (x|λ)= 1 1 − λ ( H n ( 2 ) ( x | λ ) − H n ( x | λ ) ) .$
(19)

For $n∈ Z +$, let $p(x)∈ P n$. Then we note that

$(1−λ)p(x)= ∑ k = 0 n 1 k ! { p ( k ) ( 1 ) − λ p ( k ) ( 0 ) } H k (x|λ)(see ).$
(20)

Let us take $p(x)= H n (x)$. Then, by (20), we get

$( 1 − λ ) H n ( x ) = ∑ k = 0 n 1 k ! { 2 k n ! ( n − k ) ! H n − k ( 1 ) − λ 2 k n ! ( n − k ) ! H n − k } H k ( x | λ ) = ∑ k = 0 n ( n k ) 2 k ( H n − k ( 1 ) − λ H n − k ) H k ( x | λ ) (see ).$
(21)

Therefore, by (21), we obtain the following theorem.

Theorem 12 For $n∈ Z +$, we have

$(1−λ) H n (x)= ∑ k = 0 n ( n k ) 2 k ( H n − k ( 1 ) − λ H n − k ) H k (x|λ).$

Let us take $d d λ$ on the both sides of Theorem 12.

Then, we have

$− H n ( x ) = − ∑ k = 0 n ( n k ) 2 k H n − k H k ( x | λ ) + ∑ k = 0 n ( n k ) 2 k ( H n − k ( 1 ) − λ H n − k ) ( d d λ H k ( x | λ ) ) .$
(22)

By (22), we get

$H n ( x ) = ∑ k = 0 n ( n k ) 2 k H n − k H k ( x | λ ) + ∑ k = 0 n ( n k ) ( λ H n − k − H n − k ( 1 ) ) 2 k ( d d λ H k ( x | λ ) ) .$
(23)

References

1. 1.

Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399–406.

2. 2.

Acikgoz M, Erdal D, Araci S: A new approach to q -Bernoulli numbers and q -Bernoulli polynomials related to q -Bernstein polynomials. Adv. Differ. Equ. 2010., 2010: Article ID 951764

3. 3.

Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133–143. 10.1134/S1061920811020014

4. 4.

Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler L -functions. Adv. Stud. Contemp. Math. 2009, 18(2):135–160.

5. 5.

Carlitz L: A note on q -Eulerian numbers. J. Comb. Theory, Ser. A 1978, 25(1):90–94. 10.1016/0097-3165(78)90038-9

6. 6.

Cangul IN, Simsek Y: A note on interpolation functions of the Frobenius-Euler numbers. AIP Conf. Proc. 1301. In Application of Mathematics in Technical and Natural Sciences. Am. Inst. Phys., Melville; 2010:59–67.

7. 7.

Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on $Z p$ . Adv. Stud. Contemp. Math. 2012, 22(2):215–223.

8. 8.

Kim DS, Kim T: A note on higher-order Bernoulli polynomials. J. Inequal. Appl. 2013., 2013: Article ID 111

9. 9.

Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on $Z p$ . Rocky Mt. J. Math. 2011, 41(1):239–247. 10.1216/RMJ-2011-41-1-239

10. 10.

Kim DS, Kim T: Some identities of higher-order Euler polynomials arising from Euler basis. Integral Transforms Spec. Funct. 2013. doi:10.1080/10652469.2012.754756

11. 11.

Kim DS, Kim T: Some new identities of Frobenius-Euler numbers and polynomials. J. Inequal. Appl. 2012., 2012: Article ID 307. doi:10.1186/1029–242X-2012–307

12. 12.

Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(12):2854–2865. 10.1016/j.jnt.2012.05.033

13. 13.

Kim T: New approach to q -Euler polynomials of higher order. Russ. J. Math. Phys. 2010, 17(2):218–225. 10.1134/S1061920810020068

14. 14.

Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on $Z p$ . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037

15. 15.

Kim T, Choi J, Kim YH: On q -Bernstein and q -Hermite polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):215–221.

16. 16.

Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.

17. 17.

Ryoo CS, Agarwal RP: Calculating zeros of the twisted Genocchi polynomials. Adv. Stud. Contemp. Math. 2008, 17(2):147–159.

18. 18.

Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math. 2007, 15(2):187–194.

19. 19.

Shiratani K: On the Euler numbers. Mem. Fac. Sci., Kyushu Univ., Ser. A 1973, 27: 1–5.

Acknowledgements

This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

Author information

Correspondence to Taekyun Kim.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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