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New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials

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Abstract

In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and recurrence relations between these polynomials in (Mahmudov in Discrete Dyn. Nat. Soc. 2012:169348, 2012; Mahmudov in Adv. Differ. Equ. 2013:1, 2013).

MSC:05A10, 11B65, 28B99, 11B68.

1 Introduction, definitions and notations

In the usual notations, let B n (x) and E n (x) denote, respectively, the classical Bernoulli and Euler polynomials of degree n in x, defined by the generating functions

n = 0 B n (x) t n n ! = t e t 1 e x t ,|t|<2π

and

n = 0 E n (x) t n n ! = 2 e t + 1 e x t ,|t|<π.

Also, let

B n := B n (0)and E n := E n (0),

where B n and E n are, respectively, the Bernoulli and Euler numbers of order n.

Carlitz first extended the classical Bernoulli polynomials and numbers, Euler polynomials and numbers [1]. There are numerous recent investigations on this subject by many authors. Cheon [2], Kurt [3], Luo [4], Luo and Srivastava [5], Srivastava et al. [6, 7], Tremblay et al. [8], and Mahmudov [9, 10].

Throughout this paper, we always make use of the following notation: denotes the set of natural numbers and denotes the set of complex numbers.

The q-numbers and q-factorial are defined by

[ a ] q = 1 q a 1 q ,q1, [ n ] q != [ n ] q [ n 1 ] q [ 2 ] q [ 1 ] q ,nN,aC,

respectively, where [ 0 ] q !=1, nN, aC. The q-polynomials coefficient is defined by

[ n k ] q = ( q : q ) n ( q : q ) n k ( q : q ) k ,

where ( q : q ) n =(1q) ( 1 q n ) n .

The q-analogue of the function ( x + y ) q n is defined by

( x + y ) q n = k = 0 n [ n k ] q q k ( k 1 ) 2 x n k y k .

The q-binomial formula is known as

( n : q ) a = ( 1 a ) q n = j = 0 n 1 ( 1 q j a ) = k = 0 n [ n k ] q q k ( k 1 ) 2 ( 1 ) k a k .

The q-exponential functions are given by

e q (z)= n = 0 z n [ n ] q ! = k = 0 1 ( 1 ( 1 q ) q k z ) ,0<|q|<1,|z|< 1 | 1 q |

and

E q (z)= n = 0 q n ( n 1 ) 2 z n [ n ] q ! = k = 0 ( 1 + ( 1 q ) q k z ) ,0<|q|<1,zC.

From these forms, we easily see that e q (z) E q (z)=1. Moreover, D q e q (z)= e q (z), D q E q (z)= E q (qz), where D q is defined by

D q f(z)= f ( q z ) f ( z ) q z z ,0<|q|<1,0zC.

The above q-standard notation can be found in [10].

Mahmudov defined and studied properties of the following generalized q-Bernoulli polynomials B n , q ( α ) (x,y) of order α and q-Euler polynomials E n , q ( α ) (x,y) of order α as follows [10].

Let qC, αN and 0<|q|<1. The q-Bernoulli numbers B n , q ( α ) and polynomials B n , q ( α ) (x,y) in x, y of order α are defined by means of the generating functions

n = 0 B n , q ( α ) t n [ n ] q ! = ( t e q ( t ) 1 ) α ,|t|<2π,
(1)
n = 0 B n , q ( α ) (x,y) t n [ n ] q ! = ( t e q ( t ) 1 ) α e q (tx) E q (ty),|t|<2π.
(2)

The q-Euler numbers E n , q ( α ) and polynomials E n , q ( α ) (x,y) in x, y of order α are defined by means of the generating functions

n = 0 E n , q ( α ) t n [ n ] q ! = ( 2 e q ( t ) + 1 ) α ,|t|<π,
(3)
n = 0 E n , q ( α ) (x,y) t n [ n ] q ! = ( 2 e q ( t ) + 1 ) α e q (tx) E q (ty),|t|<π.
(4)

The q-Genocchi numbers G n , q ( α ) and polynomials G n , q ( α ) (x,y) in x, y of order α are defined by means of the generating functions

n = 0 G n , q ( α ) t n [ n ] q ! = ( 2 t e q ( t ) + 1 ) α ,|t|<π,
(5)
n = 0 G n , q ( α ) (x,y) t n [ n ] q ! = ( 2 t e q ( t ) + 1 ) α e q (tx) E q (ty),|t|<π.
(6)

It is obvious that

B n , q ( α ) = B n , q ( α ) ( 0 , 0 ) , lim q 1 B n , q ( α ) ( x , y ) = B n ( α ) ( x + y ) , lim q 1 B n , q ( α ) = B n ( α ) , E n , q ( α ) = E n , q ( α ) ( 0 , 0 ) , lim q 1 E n , q ( α ) ( x , y ) = E n ( α ) ( x + y ) , lim q 1 E n , q ( α ) = E n ( α )

and

G n , q ( α ) = G n , q ( α ) (0,0), lim q 1 G n , q ( α ) (x,y)= G n ( α ) (x+y), lim q 1 G n , q ( α ) = G n ( α ) .

From (2), (4) and (6), it is easy to check that

B n , q ( α ) ( x , y ) = k = 0 n [ n k ] q B n k , q ( x , 0 ) B k , q ( α 1 ) ( 0 , y ) , E n , q ( α ) ( x , y ) = k = 0 n [ n k ] q E n k , q ( x , 0 ) E k , q ( α 1 ) ( 0 , y )

and

G n , q ( α ) (x,y)= k = 0 n [ n k ] q G n k , q (x,0) G k , q ( α 1 ) (0,y).

In this work, we give a different form of the analogue of the Srivastava-Pintér addition theorem.

More precisely, we prove

G n , q ( x , y ) = y G n 1 , q ( x , q y ) + x G n 1 , q ( x , y ) G n , q ( x , y ) = + 1 [ n ] q { G n , q ( x , y ) 1 2 k = 0 n [ n k ] q G k , q ( x , y ) G n k , q ( 1 , 0 ) } , k = 0 n [ n k ] q G k , q ( x , y ) + G n , q ( x , y ) = 2 [ n ] q ( x + y ) q n 1 , G n , q ( α ) ( x , y ) = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j k + G k , q ( α ) ( x , 0 ) } G n + 1 k , q ( 0 , m y ) m k n = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( 0 , y ) m j k + G k + 1 , q ( α ) ( 0 , y ) } × G n + 1 k , q ( m x , 0 ) m k n , G n , q ( α ) ( x , y ) = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j n G k , q ( α ) ( x , 0 ) } B n + 1 k , q ( 0 , m y ) m k n , B n , q ( α ) ( x , y ) = 1 2 r = 0 n + 1 [ n + 1 r ] q 1 [ n + 1 ] q ( r = 0 k [ k r ] q B k , q ( α ) ( x , 0 ) m k r + B r , q ( α ) ( x , 0 ) ) × G n + 1 r , q ( 0 , m y ) m r n .

2 Main theorems

Proposition 2.1 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

k = 0 n [ n k ] q B k , q ( α ) (x,0) B n k , q ( α ) = x n ,
(7)
k = 0 n [ n k ] q B k , q ( α ) (0,y) B n k , q ( α ) = q n ( n 1 ) 2 y n ,
(8)
B n , q ( α ) (x,y)= l = 0 n [ n l ] q B n l , q ( α ) (0,y) k = 0 l [ l k ] q E k , q ( α ) (x,0) E l k , q ( α ) (0,0),
(9)
E n , q ( α ) (x,y)= l = 0 n [ n l ] q E n l , q ( α ) (0,y) k = 0 l [ l k ] q E k , q ( α ) (x,0) B l k , q ( α ) (0,0).
(10)

Proposition 2.2 For x,y,zC, the following relations hold true:

G n , q ( α ) (x+z,y)= p = 0 n [ n p ] q G n p , q ( α ) (0,y) r = 0 p [ p r ] q x r z p r ,
(11)
k = 0 n [ n k ] q G k , q ( α ) (x,y) G n k , q ( α ) (0,0)= k = 0 n [ n k ] q x k y n k q ( n k ) ( n k 1 ) 2 = ( x + y ) q n .
(12)

Proof The proof of these propositions can be found from (1)-(6). □

Theorem 2.3 The generalized q-Genocchi polynomials satisfy the following recurrence relation:

G n , q ( x , y ) = y G n 1 , q ( x , q y ) + x G n 1 , q ( x , y ) + 1 [ n ] q { G n , q ( x , y ) 1 2 k = 0 n [ n k ] q G k , q ( x , y ) G n k , q ( 1 , 0 ) } .
(13)

Proof In (6) for α=1, we take the q-derivative of the generalized q-Genocchi polynomials G n , q (x,y) according to t. We note that

n = 0 D q , t G n , q ( x , y ) t n [ n ] q ! = D q , t { 2 t e q ( t ) + 1 e q ( t x ) E q ( y t ) } = 2 e q ( t x ) E q ( y t ) e q ( t ) + 1 + y 2 t e q ( t x ) E q ( y t ) e q ( t ) + 1 + x 2 t e q ( t x ) E q ( y t ) e q ( t ) + 1 2 t e q ( t x ) E q ( y t ) e q ( t ) + 1 e q ( x ) e q ( t ) + 1

and

n = 0 G n + 1 , q ( x , y ) t n [ n ] q ! = 1 t n = 0 G n , q ( x , y ) t n [ n ] q ! + y n = 0 G n , q ( x , q y ) t n [ n ] q ! + x n = 0 G n , q ( x , y ) t n [ n ] q ! 1 2 t n = 0 G n , q ( x , y ) t n [ n ] q ! n = 0 G n , q ( 1 , 0 ) t n [ n ] q ! .

If we take necessary operation, comparing the coefficients of t n [ n ] q ! , we have (13). □

Theorem 2.4 There is the following relation for the q-Genocchi polynomials:

k = 0 n [ n k ] q ( G k , q ( α ) ( x , 0 ) + G k , q ( α ) ( x , 1 ) ) =2 [ n ] q G n 1 , q ( α 1 ) (x,0).
(14)

Proof From (6) and e q (z) E q (z)=1, we have

n = 0 G n , q ( α ) (x,0) t n [ n ] q ! + n = 0 G n , q ( α ) (x,1) t n [ n ] q ! = ( 2 t e q ( t ) + 1 ) α e q (tx) ( 1 + E q ( t ) )

and

n = 0 ( G n , q ( α ) ( x , 0 ) + G n , q ( α ) ( x , 1 ) ) t n [ n ] q ! =2t n = 0 G n , q ( α 1 ) (x,0) t n [ n ] q ! .

Thus, we obtain

n = 0 { k = 0 n [ n k ] q ( G k , q ( α ) ( x , 0 ) + G k , q ( α ) ( x , 1 ) ) } t n [ n ] q ! =2 n = 1 [ n ] q G n 1 , q ( α 1 ) (x,0) t n [ n ] q ! .

From this last equality, we have (14). □

Theorem 2.5 There is the following identity for the q-Genocchi polynomials:

k = 0 n [ n k ] q G k , q (x,y)+ G n , q (x,y)=2 [ n ] q ( x + y ) q n 1 .
(15)

Proof From e q (t) E q (t)=1, we write as

1 E q ( t ) + 1 = 1 1 e q ( t ) + 1 , 2 t e q ( t x ) E q ( y t ) E q ( t ) + 1 = 2 t e q ( t x ) E q ( y t ) 2 t e q ( t x ) E q ( y t ) e q ( t ) + 1 , 2 t e q ( t ) + 1 e q ( t x ) E q ( y t ) e q ( t ) = 2 t e q ( t x ) E q ( t y ) n = 0 G n , q ( x , y ) t n [ n ] q ! , n = 0 G n , q ( x , y ) t n [ n ] q ! n = 0 t n [ n ] q ! = 2 n = 0 ( x , y ) q n t n + 1 [ n ] q ! n = 0 G n , q ( x , y ) t n [ n ] q ! .

By using the Cauchy product, compression of the results, we have (15). □

Theorem 2.6 There are the following relationships for the q-Genocchi polynomials:

G n , q ( α ) ( x , y ) = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j k + G k , q ( α ) ( x , 0 ) } × G n + 1 k , q ( 0 , m y ) m k n ,
(16)
G n , q ( α ) ( x , y ) = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( 0 , y ) m j k + G k + 1 , q ( α ) ( 0 , y ) } × G n + 1 k , q ( m x , 0 ) m k n .
(17)

Proof Proof of (16), we write

n = 0 G n , q ( α ) ( x , y ) t n [ n ] q ! = ( 2 t e q ( t ) + 1 ) α e q ( t x ) E q ( t y ) = ( 2 t e q ( t ) + 1 ) α e q ( t x ) e q ( t m ) + 1 t m t m e q ( t m ) + 1 = m t { n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! n = 0 t n m n [ n ] q ! + n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! } × n = 0 G n , q ( 0 , m y ) t n m n [ n ] q ! = n = 0 ( 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j k + G k , q ( α ) ( x , 0 ) } × G n + 1 k , q ( 0 , m y ) m k n ) t n [ n ] q ! .

Comparing the coefficients of t n [ n ] q ! , we have (16). The proof of (17) is similar to that of (16). □

3 Explicit relation between the q-Bernoulli polynomials and q-Genocchi polynomials

In this section, we prove two interesting relations between the q-Bernoulli polynomials B n , q ( α ) (x,y) of order α and the q-Genocchi polynomials G n , q ( α ) (x,y) of order α.

Theorem 3.1 There is the following relation between q-Genocchi polynomials and q-Bernoulli polynomials

G n , q ( α ) ( x , y ) = 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j n G k , q ( α ) ( x , 0 ) } × B n + 1 k , q ( 0 , m y ) m k n .
(18)

Proof From (6), we deduce that

n = 0 G n , q ( α ) ( x , y ) t n [ n ] q ! = ( 2 t e q ( t ) + 1 ) α e q ( t x ) E q ( t y ) = m t { n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! n = 0 t n m n [ n ] q ! n = 0 B n , q ( 0 , m y ) t n m n [ n ] q ! n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! n = 0 B n , q ( 0 , m y ) t n m n [ n ] q ! } = m t { n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! n = 0 t n m n [ n ] q ! n = 0 G n , q ( α ) ( x , 0 ) t n [ n ] q ! } × n = 0 B n , q ( 0 , m y ) t n m n [ n ] q ! = n = 0 ( 1 [ n + 1 ] q k = 0 n + 1 [ n + 1 k ] q { j = 0 k [ k j ] q G j , q ( α ) ( x , 0 ) m j n G k , q ( α ) ( x , 0 ) } × B n + 1 k , q ( 0 , m y ) m k n ) t n [ n ] q ! .

Comparing the coefficients of t n [ n ] q ! , we have (18). □

Theorem 3.2 There is the following relation between q-Bernoulli polynomials and q-Genocchi polynomials:

B n , q ( α ) ( x , y ) = 1 2 r = 0 n + 1 [ n + 1 r ] q 1 [ n + 1 ] q ( r = 0 k [ k r ] q B k , q ( α ) ( x , 0 ) m k r + B r , q ( α ) ( x , 0 ) ) × G n + 1 r , q ( 0 , m y ) m r n .
(19)

Proof From (2), we obtain

n = 0 B n , q ( α ) ( x , y ) t n [ n ] q ! = ( t e q ( t ) 1 ) α e q ( t x ) E q ( t y ) = m 2 t { ( t e q ( t ) 1 ) α e q ( t x ) e q ( t m ) 2 t m e q ( t m ) + 1 E q ( t m , m y ) + ( t e q ( t ) 1 ) α e q ( t x ) 2 t m e q ( t m ) + 1 E q ( t m , m y ) } = m 2 t { n = 0 B n , q ( α ) ( x , 0 ) t n [ n ] q ! n = 0 t n m n [ n ] q ! + n = 0 B n , q ( α ) ( x , 0 ) t n [ n ] q ! } × n = 0 G n , q ( 0 , m y ) t n m n [ n ] q ! = m 2 n = 0 r = 0 n [ n r ] q ( r = 0 k [ k r ] q B k , q ( α ) ( x , 0 ) m k r + B r , q ( α ) ( x , 0 ) ) × G n r , q ( 0 , m y ) m r n 1 [ n ] q t n 1 [ n 1 ] q ! = m 2 n = 1 { 1 2 r = 0 n + 1 [ n + 1 r ] q 1 [ n + 1 ] q × ( r = 0 k [ k r ] q B k , q ( α ) ( x , 0 ) m k r + B r , q ( α ) ( x , 0 ) ) × G n + 1 r , q ( 0 , m y ) m r n } t n [ n ] q ! .

Comparing the coefficients of t n [ n ] q ! , we have (19). □

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Acknowledgements

This paper was supported by the Scientific Research Project Administration of Akdeniz University. The author is grateful to the referees for valuable comments. Proceedings of 2nd International Eurasian Conference on Mathematical Sciences and Applications.

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Kurt, V. New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials. Adv Differ Equ 2014, 5 (2014) doi:10.1186/1687-1847-2014-5

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Keywords

  • Bernoulli numbers and polynomials
  • Genocchi polynomials
  • generating function
  • generalized Bernoulli polynomials
  • generalized Genocchi polynomials
  • q-Bernoulli polynomials
  • q-Genocchi polynomials