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On a class of generalized q-Bernoulli and q-Euler polynomials

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Abstract

The main purpose of this paper is to introduce and investigate a new class of generalized q-Bernoulli and q-Euler polynomials. The q-analogues of well-known formulas are derived. A generalization of the Srivastava-Pintér addition theorem is obtained.

1 Introduction

Throughout this paper, we always make use of the following notation: denotes the set of natural numbers, N 0 denotes the set of nonnegative integers, denotes the set of real numbers, denotes the set of complex numbers.

The q-numbers and q-factorial are defined by

[ a ] q = 1 q a 1 q (q1); [ 0 ] q !=1; [ n ] q != [ 1 ] q [ 2 ] q [ n ] q ,nN,aC,

respectively. The q-polynomial coefficient is defined by

[ n k ] q = ( q ; q ) n ( q ; q ) n k ( q ; q ) k .

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

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

The q-binomial formula is known as

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

In the standard approach to the q-calculus, two exponential functions are used:

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

From this form, 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 [1].

Carlitz firstly extended the classical Bernoulli and Euler numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials [24]. There are numerous recent investigations on this subject by, among many other authors, Cenki et al. [57], Choi et al. [8] and [9], Kim et al. [1013], Ozden and Simsek [14], Ryoo et al. [15], Simsek [16, 17] and [18], and Luo and Srivastava [19], Srivastava et al. [20], Mahmudov [21, 22].

Recently, Natalini and Bernardini [23], Bretti et al. [24], Kurt [25, 26], Tremblay et al. [27, 28] studied the properties of the following generalized Bernoulli and Euler polynomials:

( t m e t k = 0 m 1 t k k ! ) α e t x = n = 0 B n [ m 1 , α ] ( x ) t n n ! , ( t m e t + k = 0 m 1 t k k ! ) α e t x = n = 0 E n [ m 1 , α ] ( x ) t n n ! , α C , 1 α : = 1 .
(1)

Motivated by the generalizations in (1) of the classical Bernoulli and Euler polynomials, we introduce and investigate here the so-called generalized two-dimensional q-Bernoulli and q-Euler polynomials, which are defined as follows.

Definition 1 Let q,αC, mN, 0<|q|<1. The generalized two-dimensional q-Bernoulli polynomials B n , q [ m 1 , α ] (x,y) are defined, in a suitable neighborhood of t=0, by means of the generating function

( t m e q ( t ) T m 1 , q ( t ) ) α e q (tx) E q (ty)= n = 0 B n , q [ m 1 , α ] (x,y) t n [ n ] q ! ,

where T m 1 , q (t)= k = 0 m 1 t k [ k ] q ! .

Definition 2 Let q,αC, 0<|q|<1, mN. The generalized two-dimensional q-Euler polynomials E n , q [ m 1 , α ] (x,y) are defined, in a suitable neighborhood of t=0, by means of the generating functions

( 2 m e q ( t ) + T m 1 , q ( t ) ) α e q (tx) E q (ty)= n = 0 E n , q [ m 1 , α ] (x,y) t n [ n ] q ! .

It is obvious that

lim q 1 B n , q [ m 1 , α ] ( x , y ) = B n [ m 1 , α ] ( x + y ) , B n , q [ m 1 , α ] = B n , q [ m 1 , α ] ( 0 , 0 ) , lim q 1 B n , q [ m 1 , α ] = B n [ m 1 , α ] , lim q 1 E n , q [ m 1 , α ] ( x , y ) = E n [ m 1 , α ] ( x + y ) , E n , q [ m 1 , α ] = E n , q [ m 1 , α ] ( 0 , 0 ) , lim q 1 E n , q [ m 1 , α ] = E n [ m 1 , α ] , lim q 1 B n , q [ m 1 , α ] ( x , 0 ) = B n [ m 1 , α ] ( x ) , lim q 1 B n , q [ m 1 , α ] ( 0 , y ) = B n [ m 1 , α ] ( y ) , lim q 1 E n , q [ m 1 , α ] ( x , 0 ) = E n [ m 1 , α ] ( x ) , lim q 1 E n , q [ m 1 , α ] ( 0 , y ) = E n [ m 1 , α ] ( y ) .

Here B n [ m 1 , α ] (x) and E n [ m 1 , α ] (x) denote the generalized Bernoulli and Euler polynomials defined in (1). Notice that B n [ m 1 , α ] (x) was introduced by Natalini [23], and E n [ m 1 , α ] (x) was introduced by Kurt [25].

In fact Definitions 1 and 2 define two different types B n , q [ m 1 , α ] (x,0) and B n , q [ m 1 , α ] (0,y) of the generalized q-Bernoulli polynomials and two different types E n , q [ m 1 , α ] (x,0) and E n , q [ m 1 , α ] (0,y) of the generalized q-Euler polynomials. Both polynomials B n , q [ m 1 , α ] (x,0) and B n , q [ m 1 , α ] (0,y) ( E n , q [ m 1 , α ] (x,0) and E n , q [ m 1 , α ] (0,y)) coincide with the classical higher-order generalized Bernoulli polynomials (Euler polynomials) in the limiting case q 1 .

2 Preliminaries and lemmas

In this section we provide some basic formulas for the generalized q-Bernoulli and q-Euler polynomials to obtain the main results of this paper in the next section. The following result is a q-analogue of the addition theorem for the classical Bernoulli and Euler polynomials.

Lemma 3 For all x,yC we have

B n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( x + y ) q n k , E n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( x + y ) q n k ,
(2)
B n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] ( x , 0 ) y n k = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( 0 , y ) x n k ,
(3)
E n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q [ m 1 , α ] ( x , 0 ) y n k = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( 0 , y ) x n k .
(4)

In particular, setting x=0 and y=0 in (3) and (4), we get the following formulae for the generalized q-Bernoulli and q-Euler polynomials, respectively,

B n , q [ m 1 , α ] ( x , 0 ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] x n k , B n , q [ m 1 , α ] ( 0 , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] y n k , E n , q [ m 1 , α ] ( x , 0 ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] x n k , E n , q [ m 1 , α ] ( 0 , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q [ m 1 , α ] y n k .

Setting y=1 and x=1 in (3) and (4), we get, respectively,

B n , q [ m 1 , α ] ( x , 1 ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] ( x , 0 ) , B n , q [ m 1 , α ] ( 1 , y ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( 0 , y ) ,
(5)
E n , q [ m 1 , α ] ( x , 1 ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q ( α ) ( x , 0 ) , E n , q [ m 1 , α ] ( 1 , y ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( 0 , y ) .
(6)

Clearly, (5) and (6) are the generalization of q-analogues of

B n (x+1)= k = 0 n ( n k ) B k (x), E n (x+1)= k = 0 n ( n k ) E k (x),

respectively.

Lemma 4 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

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

Lemma 5 We have

D q , x B n , q [ m 1 , α ] ( x , y ) = [ n ] q B n 1 , q [ m 1 , α ] ( x , y ) , D q , y B n , q [ m 1 , α ] ( x , y ) = [ n ] q B n 1 , q [ m 1 , α ] ( x , q y ) , D q , x E n , q [ m 1 , α ] ( x , y ) = [ n ] q E n 1 , q [ m 1 , α ] ( x , y ) , D q , y E n , q [ m 1 , α ] ( x , y ) = [ n ] q E n 1 , q [ m 1 , α ] ( x , q y ) .

Lemma 6 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) = [ n ] q ! [ n m ] q ! B n m , q [ m 1 , α 1 ] ( 0 , y ) , n m , E n , q [ m 1 , α ] ( 1 , y ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , α ] ( 0 , y ) = 2 m E n , q [ m 1 , α 1 ] ( 0 , y ) , B n , q [ m 1 , α ] ( x , 0 ) k = 0 min ( n , m 1 ) [ n k ] q B n , q [ m 1 , α ] ( x , 1 ) = [ n ] q ! [ n m ] q ! B n m , q [ m 1 , α 1 ] ( x , 1 ) , n m , E n , q [ m 1 , α ] ( x , 0 ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , α ] ( x , 1 ) = 2 m E n , q [ m 1 , α 1 ] ( x , 1 ) .
(7)

Proof We prove only (7). The proof is based on the following equality:

n = 0 ( B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) T m 1 , q ( t ) ) α e q ( t ) E q ( t y ) T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) ( e q ( t ) T m 1 , q ( t ) ) = t m ( t m e q ( t ) T m 1 , q ( t ) ) α 1 E q ( t y ) = n = 0 [ n + m ] q ! [ n ] q ! B n , q [ m 1 , α 1 ] ( 0 , y ) t n + m [ n + m ] q ! .

Here we used the following relation:

T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = n = 0 m 1 t n [ n ] q ! n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = n = 0 B n , q [ m 1 , α ] ( 0 , y ) ( t n [ n ] q ! + t n + 1 [ n ] q ! + t n + 2 [ n ] q ! [ 2 ] q ! + + t n + m 1 [ n ] q ! [ m 1 ] q ! ) = n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + n = 0 [ n ] q B n 1 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + n = 0 [ n ] q [ n 1 ] q [ 2 ] q ! B n 2 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + + n = 0 [ n ] q [ n m + 2 ] q [ m 1 ] q ! B n m + 1 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = n = 0 k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! .

 □

Corollary 7 Taking q 1 , we have

B n [ m 1 , α ] ( y + 1 ) k = 0 min ( n , m 1 ) [ n k ] q B n k [ m 1 , α ] ( y ) = [ n ] q ! [ n m ] q ! B n m [ m 1 , α 1 ] ( y ) , n m , E n [ m 1 , α ] ( y + 1 ) + k = 0 min ( n , m 1 ) [ n k ] q E n [ m 1 , α ] ( y ) = 2 m E n [ m 1 , α 1 ] ( y ) .

Lemma 8 The generalized q-Bernoulli polynomials satisfy the following relations:

B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) = [ n ] q k = 0 n 1 [ n 1 k ] q B k , q [ m 1 , α ] ( 0 , y ) B n 1 k , q [ 0 , 1 ] .
(8)

Proof

Indeed,

n = 0 ( B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) T m 1 , q ( t ) ) α e q ( t ) E q ( t y ) T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) e q ( t ) T m 1 , q ( t ) t t = n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! n = 0 B n , q [ 0 , 1 ] t n + 1 [ n ] q ! = n = 1 [ n ] q k = 0 n 1 [ n 1 k ] q B k , q [ m 1 , α ] ( 0 , y ) B n 1 k , q [ 0 , 1 ] t n [ n ] q ! .

 □

Remark 9 Notice taking limit in (8) as q 1 , we get

B n [ m 1 , α ] (y+1) k = 0 min ( n , m 1 ) ( n k ) B n k [ m 1 , α ] (y)=n k = 0 n 1 ( n 1 k ) B k [ m 1 , α ] (y) B n 1 k [ 0 , 1 ] .

It is a correct form of formula (2.7) from [27] for λ=1.

Lemma 10 We have

x n = k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n k , q [ m 1 , 1 ] ( x , 0 ) , y n = 1 q n ( n 1 ) 2 k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n k , q [ m 1 , 1 ] ( 0 , y ) , x n = 1 2 m ( k = 0 n [ n k ] q E k , q [ m 1 , 1 ] ( x , 0 ) + k = 0 min ( n , m 1 ) [ n k ] q E k , q [ m 1 , 1 ] ( x , 0 ) ) , y n = 1 2 m q n ( n 1 ) 2 ( k = 0 n [ n k ] q E k , q [ m 1 , 1 ] ( 0 , y ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , 1 ] ( 0 , y ) ) .

From Lemma 10 we obtain the list of generalized q-Bernoulli polynomials as follows

B 0 , q [ m 1 , 1 ] ( x , 0 ) = [ m ] q ! , B 0 , q [ m 1 , 1 ] ( 0 , y ) = [ m ] q ! , B 1 , q [ m 1 , 1 ] ( x , 0 ) = [ m ] q ! ( x 1 [ m + 1 ] q ) , B 1 , q [ m 1 , 1 ] ( 0 , y ) = [ m ] q ! ( y 1 [ m + 1 ] q ) , B 2 , q [ m 1 , 1 ] ( x , 0 ) = x 2 [ 2 ] q [ m ] q ! [ m + 1 ] q x + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q , B 2 , q [ m 1 , 1 ] ( 0 , y ) = q y 2 [ 2 ] q [ m ] q ! [ m + 1 ] q y + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q .

3 Explicit relationship between the q-Bernoulli and q-Euler polynomials

In this section, we give some generalizations of the Srivastava-Pintér addition theorem. We also obtain new formulae and their some special cases below.

We present natural q-extensions of the main results of the papers [29, 30].

Theorem 11 The relationships

B n , q [ m 1 , α ] ( x , y ) = 1 2 k = 0 n [ n k ] q [ 1 l n k B k , q [ m 1 , α ] ( x , 0 ) + j = 0 k [ k j ] q 1 l k j B j , q [ m 1 , α ] ( x , 0 ) ] E n k , q ( 0 , l y ) ,
(9)
B n , q [ m 1 , α ] (x,y)= 1 2 k = 0 n [ n k ] q 1 l n k [ B k , q [ m 1 , α ] ( 0 , y ) + B k , q [ m 1 , α ] ( 1 l , y ) ] E n k , q (lx,0)
(10)

hold true between the generalized q-Bernoulli polynomials and q-Euler polynomials.

Proof First we prove (9). Using the identity

( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 E q ( t l l y ) e q ( t l ) + 1 2 ( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) ,

we have

n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 E n , q ( 0 , l y ) t n l n [ n ] q ! k = 0 t k l k [ k ] q ! j = 0 B j , q [ m 1 , α ] ( x , 0 ) t j [ j ] q ! + 1 2 k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! = : I 1 + I 2 .

It is clear that

I 2 = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( x , 0 ) E n k , q ( 0 , l y ) t n [ n ] q ! .

On the other hand,

I 1 = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! j = 0 t j l j [ j ] q ! = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 j = 0 k [ k j ] q E j , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q B k , q [ m 1 , α ] ( x , 0 ) j = 0 n k [ n k j ] q 1 l n k E j , q ( 0 , l y ) t n [ n ] q ! = 1 2 n = 0 j = 0 n [ n j ] q E j , q ( 0 , l y ) k = 0 n j [ n j k ] q 1 l n k B k , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! .

Therefore

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

Next we prove (10). Using the identity

( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 e q ( t l l x ) e q ( t l ) + 1 2 ( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α E q ( t y ) ,

we have

n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 E n , q ( l x , 0 ) t n l n [ n ] q ! n = 0 B n , q [ m 1 , α ] ( 1 l , y ) t n [ n ] q ! + 1 2 k = 0 E k , q ( l x , 0 ) t k l k [ k ] q ! n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = : I 1 + I 2 .

It is clear that

I 2 = 1 2 n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! k = 0 E k , q ( l x , 0 ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( 0 , y ) E n k , q ( l x , 0 ) t n [ n ] q ! .

On the other hand,

I 1 = 1 2 n = 0 B n , q [ m 1 , α ] ( 1 l , y ) t n [ n ] q ! k = 0 E k , q ( l x , 0 ) t k m k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( 1 l , y ) E n k , q ( l x , 0 ) t n [ n ] q ! .

Therefore

n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n [ B k , q [ m 1 , α ] ( 0 , y ) + B k , q [ m 1 , α ] ( 1 l , y ) ] E n k , q ( l x , 0 ) t n [ n ] q ! .

 □

Next we discuss some special cases of Theorem 11.

Theorem 12 The relationship

B n , q [ m 1 , α ] ( x , y ) = 1 2 k = 0 n [ n k ] q [ B k , q [ m 1 , α ] ( 0 , y ) + k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) + [ k ] q j = 0 k 1 [ k 1 j ] q B j , q [ m 1 , α ] ( 0 , y ) B k 1 j , q [ 0 , 1 ] ] E n k , q ( x , 0 )

holds true between the generalized q-Bernoulli polynomials and the q-Euler polynomials.

Remark 13 Taking q 1 in Theorem 12, we obtain the Srivastava-Pintér addition theorem for the generalized Bernoulli and Euler polynomials.

B n [ m 1 , α ] ( x + y ) = 1 2 k = 0 n ( n k ) [ B k [ m 1 , α ] ( y ) + k = 0 min ( n , m 1 ) ( n k ) B n k [ m 1 , α ] ( y ) + k j = 0 k 1 ( k 1 j ) B j [ m 1 , α ] ( y ) B k 1 j [ 0 , 1 ] ] E n k ( x ) .
(11)

Notice that the Srivastava-Pintér addition theorem for the generalized Apostol-Bernoulli polynomials and the Apostol-Euler polynomials was given in [27]. The formula (11) is a correct version of Theorem 3 [27] for λ=1.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.

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Correspondence to Nazim I Mahmudov.

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The authors declare that they have no competing interests.

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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Keywords

  • Nonnegative Integer
  • Correct Form
  • Euler Number
  • Basic Formula
  • Addition Theorem