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q-Bernoulli numbers and q-Bernoulli polynomials revisited

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Abstract

This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010), some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994) (see Equation 9), some new generating functions for the q-Bernoulli numbers and polynomials are shown.

Mathematics Subject Classification (2000) 11B68, 11S40, 11S80

1. Introduction

As well-known definition, the Bernoulli polynomials are given by

t e t - 1 e x t = e B ( x ) t = n = 0 B n ( x ) t n n ! ,

(see [14]),

with usual convention about replacing B n(x) by B n (x). In the special case, x = 0, B n (0) = B n are called the n th Bernoulli numbers.

Let us assume that q with |q| < 1 as an indeterminate. The q-number is defined by

[ x ] q = 1 - q x 1 - q ,

(see [16]).

Note that lim q→1[x] q = x.

Since Carlitz brought out the concept of the q-extension of Bernoulli numbers and polynomials, many mathematicians have studied q-Bernoulli numbers and q-Bernoulli polynomials (see [1, 7, 5, 6, 812]). Recently, Acikgöz, Erdal, and Araci have studied to a new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials (see [7]). But, their generating function is unreasonable. The wrong properties are indicated by some counter-examples, and they are corrected.

It is point out that Acikgöz, Erdal and Araci's generating function for q-Bernoulli numbers and polynomials is unreasonable by counter examples, then the new generating function for the q-Bernoulli numbers and polynomials are given.

2. q-Bernoulli numbers and q-Bernoulli polynomials revisited

In this section, we perform a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. [7], some incorrect properties are revised.

Definition 1 (Acikgöz et al. [7]). For q with |q| < 1, let us define q-Bernoulli polynomials as follows:

D q ( t , x ) = - t y = 0 q y e [ x + y ] q t = n = 0 B n , q ( x ) t n n ! , w h e r e | t + log q | < 2 π .
(1)

In the special case, x = 0, B n,q (0) = B n,q are called the n th q-Bernoulli numbers.

Let D q (t, 0) = D q (t). Then

D q ( t ) = - t y = 0 q y e [ y ] q t = n = 0 B n , q t n n ! .
(2)

Remark 1. Definition 1 is unreasonable, since it is not the generating function of q-Bernoulli numbers and polynomials.

Indeed, by (2), we get

D q ( t , x ) = - t y = 0 q y e [ x + y ] q t = - t y = 0 q y e [ x ] q t e q x [ y ] q t (1) = - q x t q x y = 0 q y e q x [ y ] q t e [ x ] q t (2) = 1 q x e [ x ] q t D q ( q x t ) (3) = m = 0 [ x ] q m m ! t m l = 0 q ( l - 1 ) x B l , q l ! t l (4) = n = 0 l = 0 n n l [ x ] q n - l q ( l - 1 ) x B l , q t n n ! . (5) (6)
(3)

By comparing the coefficients on the both sides of (1) and (3), we obtain the following equation

B n , q ( x ) = l = 0 n n l [ x ] q n - l q ( l - 1 ) x B l , q .
(4)

From (1), we note that

D q ( t , x ) = - t y = 0 q y e [ x + y ] q t (1) = n = 0 - t y = 0 q y [ x + y ] q n t n n ! (2) = - n = 0 n + 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l x y = 0 q ( l + 1 ) y t n + 1 ( n + 1 ) ! (3) = n = 1 - n ( 1 - q ) n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q l x 1 1 - q l + 1 t n n ! . (4) (5)
(5)

By comparing the coefficients on the both sides of (1) and (5), we obtain the following equation

B 0 , q = 0 , B n , q = - n ( 1 - q ) n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q l x 1 1 - q l + 1 i f n > 0 .
(6)

By (6), we see that Definition 1 is unreasonable because we cannot derive Bernoulli numbers from Definition 1 for any q.

In particular, by (1) and (2), we get

q D q ( t , 1 ) - D q ( t ) = t .
(7)

Thus, by (7), we have

q B n , q ( 1 ) - B n , q = 1 , i f n = 1 , 0 , i f n > 1 ,
(8)

and

B n , q ( 1 ) = l = 0 n n l q l - 1 B l , q .
(9)

Therefore, by (4) and (6)-(9), we see that the following three theorems are incorrect.

Theorem 1 (Acikgöz et al. [7]). For n *, one has

B 0 , q = 1 , q ( q B + 1 ) n - B n , q = 1 , i f n = 0 , 0 , i f n > 0 .

Theorem 2 (Acikgöz et al. [7]). For n *, one has

B n , q ( x ) = l = 0 n n l q l x B l , q [ x ] q n - l .

Theorem 3 (Acikgöz et al. [7]). For n *, one has

B n , q ( x ) = 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l x l + 1 [ l + 1 ] q .

In [7], Acikgöz, Erdal and Araci derived some results by using Theorems 1-3. Hence, the other results are incorrect.

Now, we redefine the generating function of q-Bernoulli numbers and polynomials and correct its wrong properties, and rebuild the theorems of q-Bernoulli numbers and polynomials.

Redefinition 1. For q with |q| < 1, let us define q-Bernoulli polynomials as follows:

F q ( t , x ) = - t m = 0 q 2 m + x e [ x + m ] q t + ( 1 - q ) m = 0 q m e [ x + m ] q t (1) = n = 0 β n , q ( x ) t n n ! , w h e r e | t + log q | < 2 π . (2) (3)
(10)

In the special case, x = 0, β n,q (0) = β n,q are called the n th q-Bernoulli numbers.

Let F q (t, 0) = F q (t). Then we have

F q ( t ) = n = 0 β n , q t n n ! (1) = - t m = 0 q 2 m e [ m ] q t + ( 1 - q ) m = 0 q m e [ m ] q t . (2) (3)
(11)

By (10), we get

β n , q ( x ) = - n m = 0 q 2 m + x [ x + m ] q n - 1 + ( 1 - q ) m = 0 q m [ x + m ] q n (1)  = - n ( 1 - q ) n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q ( l + 1 ) x ( 1 - q l + 2 ) + ( 1 - q ) m = 0 q m [ x + m ] q n (2)  = 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l x l + 1 [ l + 1 ] q . (3)  (4) 
(12)

By (10) and (11), we get

F q ( t , x ) = e [ x ] q t F q ( q x t ) (1) = m = 0 [ x ] q m t m m ! l = 0 β l , q l ! q l x t l (2) = n = 0 l = 0 n q l x β l , q [ x ] q n - l n ! l ! ( n - l ) ! t n n ! (3) = n = 0 l = 0 n n l q l x β l , q [ x ] q n - l t n n ! . (4) (5)
(13)

Thus, by (12) and (13), we have

β n , q ( x ) = l = 0 n n l q l x β l , q [ x ] q n - l (1) = - n m = 0 q m [ x + m ] q n - 1 + ( 1 - q ) ( n + 1 ) m = 0 q m [ x + m ] q n . (2) (3)
(14)

From (10) and (11), we can derive the following equation:

q F q ( t , 1 ) - F q ( t ) = t + ( q - 1 ) .
(15)

By (15), we get

q β n , q ( 1 ) - β n , q = q - 1 , i f n = 0 , 1 , i f n = 1 , 0 i f n > 1 .
(16)

Therefore, by (14) and (15), we obtain

β 0 , q = 1 , q ( q β q + 1 ) n - β n , q = 1 , i f n = 1 , 0 i f n > 1 ,
(17)

with the usual convention about replacing β q n by β n,q .

From (12), (14) and (16), Theorems 1-3 are revised by the following Theorems 1'-3'.

Theorem 1'. For n +, we have

β 0 , q = 1 , a n d q ( q β q + 1 ) n - β n , q = 1 , i f n = 1 , 0 i f n > 1 .

Theorem 2'. For n +, we have

β n , q ( x ) = l = 0 n n l q l x β l , q [ x ] q n - l .

Theorem 3'. For n +, we have

β n , q ( x ) = 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l x l + 1 [ l + 1 ] q .

From (10), we note that

F q ( t , x ) = 1 [ d ] q a = 0 d - 1 q a F q d [ d ] q t , x + a d , d .
(18)

Thus, by (10) and (18), we have

β n , q ( x ) = [ d ] q n - 1 a = 0 d - 1 q a β n , q d x + a d , n + .

For d , let χ be Dirichlet's character with conductor d. Then, we consider the generalized q-Bernoulli polynomials attached to χ as follows:

F q , χ ( t , x ) = - t m = 0 χ ( m ) q 2 m + x e [ x + m ] q t + ( 1 - q ) m = 0 χ ( m ) q m e [ x + m ] q t (1) = n = 0 β n , χ , q ( x ) t n n ! . (2) (3)

In the special case, x = 0, β n,χ,q (0) = β n,χ,q are called the n th generalized Carlitz q-Bernoulli numbers attached to χ (see [8]).

Let F q,χ (t, 0) = F q,χ (t). Then we have

F q , χ ( t ) = - t m = 0 χ ( m ) q 2 m e [ m ] q t + ( 1 - q ) m = 0 χ ( m ) q m e [ m ] q t (1) = n = 0 β n , χ , q t n n ! . (2) (3)
(20)

From (20), we note that

β n , χ , q = - n m = 0 q 2 m χ ( m ) [ m ] q n - 1 + ( 1 - q ) m = 0 q m χ ( m ) [ m ] q n (1) = - n a = 0 d - 1 m = 0 q 2 a + 2 d m χ ( a + d m ) [ a + d m ] q n - 1 (2) + a = 0 d - 1 m = 0 q a + d m χ ( a + d m ) [ a + d m ] q n (3) = a = 0 d - 1 χ ( a ) q a - n ( 1 - q ) n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q ( l + 1 ) a ( 1 - q d ( l + 2 ) ) (4) + ( 1 - q ) a = 0 d - 1 χ ( a ) q a 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l a ( 1 - q d ( l + 1 ) ) (5) = a = 0 d - 1 χ ( a ) q a - n ( 1 - q ) n - 1 l = 0 n - 1 n - 1 l ( - 1 ) l q ( l + 1 ) a ( 1 - q d ( l + 2 ) ) (6) + a = 0 d - 1 χ ( a ) q a 1 ( 1 - q ) n - 1 l = 0 n n l ( - 1 ) l q l a ( 1 - q d ( l + 1 ) ) (7) = a = 0 d - 1 χ ( a ) q a 1 ( 1 - q ) n - 1 l = 0 n n l ( - 1 ) l q l a l ( 1 - q d ( l + 1 ) ) (8) + a = 0 d - 1 χ ( a ) q a 1 ( 1 - q ) n - 1 l = 0 n n l ( - 1 ) l q l a ( 1 - q d ( l + 1 ) ) (9) = a = 0 d - 1 χ ( a ) q a 1 - q ( 1 - q ) n l = 0 n n l ( - 1 ) l q l a l + 1 1 - q d ( l + 1 ) . (10) (11)

Therefore, by (20) and (21), we obtain the following theorem.

Theorem 4. For n +, we have

β n , χ , q = a = 0 d - 1 χ ( a ) q a 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l a l + 1 [ d ( l + 1 ) ] q (1) = - n m = 0 χ ( m ) q m [ m ] q n - 1 + ( 1 - q ) ( 1 + n ) m = 0 χ ( m ) q m [ m ] q n , (2) (3)

and

β n , χ , q ( x ) = - n m = 0 χ ( m ) q m [ m + x ] q n - 1 + ( 1 - q ) ( 1 + n ) m = 0 χ ( m ) q m [ m + x ] q n .

From (19), we note that

F q , χ ( t , x ) = 1 [ d ] q a = 0 d - 1 χ ( a ) q a F q d [ d ] q t , x + a d .
(22)

Thus, by (22), we obtain the following theorem.

Theorem 5. For n +, we have

β n , χ , q ( x ) = [ d ] q n - 1 a = 0 d - 1 χ ( a ) q a β n , q d x + a d .

For s , we now consider the Mellin transform for F q (t, x) as follows:

1 Γ ( s ) 0 F q ( - t , x ) t s - 2 d t = m = 0 q 2 m + x [ m + x ] q s + 1 - q s - 1 m = 0 q m [ m + x ] q s - 1 ,
(23)

where x ≠ 0, -1, -2,....

From (23), we note that

1 Γ ( s ) 0 F q ( - t , x ) t s - 2 d t = m = 0 q m [ m + x ] q s + ( 1 - q ) 2 - s s - 1 m = 0 q m [ m + x ] q s - 1 ,
(24)

where s , and x ≠ 0, -1, -2,....

Thus, we define q-zeta function as follows:

Definition 2. For s , q-zeta function is defined by

ζ q ( s , x ) = m = 0 q m [ m + x ] q s + ( 1 - q ) 2 - s s - 1 m = 0 q m [ m + x ] q s - 1 , R e ( s ) > 1 ,

where x ≠ 0, -1, -2,....

By (24) and Definition 2, we note that

ζ q ( 1 - n , x ) = ( - 1 ) n - 1 β n , q ( x ) n , n .

Note that

lim q 1 ζ q ( 1 - n , x ) = - B n ( x ) n ,

where B n (x) are the n th ordinary Bernoulli polynomials.

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Acknowledgements

The authors express their gratitude to the referee for his/her valuable comments.

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Correspondence to Cheon Seoung Ryoo.

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All authors contributed equally to the manuscript and read and approved the finial manuscript.

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Ryoo, C.S., Kim, T. & Lee, B. q-Bernoulli numbers and q-Bernoulli polynomials revisited. Adv Differ Equ 2011, 33 (2011). https://doi.org/10.1186/1687-1847-2011-33

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Keywords

  • Bernoulli numbers and polynomials
  • q-Bernoulli numbers and polynomials
  • q-Bernoulli numbers and polynomials