On The Degenerate Poly-Frobenius-Genocchi Polynomials Of Complex Variables
aa r X i v : . [ m a t h . N T ] J u l On The Degenerate Poly-Frobenius-GenocchiPolynomials Of Complex Variables
Burak Kurt
Akdeniz University, Mathematics of Department, Antalya TR-07058,TurkeyE-mail : [email protected]
Abstract
The main aim of this paper is to define and investigate a new class of the degenerate poly-Frobenius-Genocchi polynomials with the help of the polyexponential functions. In thispaper, we define the degenerate poly-Frobenius-Genocchi polynomials of complex variablesarising from the modified polyexponential functions. We derive explicit expressions for thesepolynomials and numbers. Also, we obtain implicit relations involving these polynomialsand some other special numbers and polynomials.
Key Words and Phrases.
Frobenius-Euler numbers and polynomials, Genocchi num-bers and polynomials, Frobenius-Genocchi numbers and polynomials, The degenerate Stir-ling numbers of both kind, The degenerate Stirling polynomials of the second kind, TheBernoulli polynomials of the second kind, The polyexponential functions, The degeneratepoly-Frobenius-Genocchi polynomials of complex variables.1.
Introduction and Notation
Throughout this paper, N denotes the set of natural numbers, N denotes the set ofnonnegative integers, R denotes the set of real numbers and C denotes the set of complexnumbers. We begin by introducing the following definitions and notations ([1]-[18]).The Frobenius-Euler polynomials H n ( x ; u ) are defined by ([1]-[18]); ∞ X n =0 H n ( x ; u ) t n n ! = 1 − ue t − u e xt , (1.1)where u = 1 and e t = u .When x = 0, H n ( u ) := H n (0; u ) are called the Frobenius-Euler numbers. Burak Kurt
The Genocchi polynomials are defined by ([11], [12], [14]) ∞ X n =0 G n ( x ) t n n ! = 2 te t + 1 e xt , | t | < π . (1.2)When x = 0, G n (0) := G n are called the Genocchi numbers.The Frobenius-Genocchi polynomials are defined by [18] ∞ X n =0 F G n ( x, u ) t n n ! = (1 − u ) te t − u e xt . (1.3)For u = − F G n ( x, −
1) = G n ( x ) and x = 0, F G n ( u ) := F G n (0 , u ) are called theFrobenius-Genocchi numbers.The degenerate exponential function is defined by ([3]-[11]) with λ ∈ R \ { } e xλ ( t ) = (1 + λt ) x/λ = ∞ X n =0 ( x ) n,λ t n n ! and e λ ( t ) = e λ ( t ) = (1 + λt ) /λ , (1.4)where ( x ) , = 1 and ( x ) n,λ = x ( x − λ ) ( x − λ ) · · · ( x − ( n − λ ), n ≥ x ∈ R and k nonnegative integer, the degenerate λ -Stirling polynomials of the secondkind are defined by [5] ( e λ ( t ) − k k ! e xλ ( t ) = ∞ X n = k S ( x )2 ,λ ( n, k ) t n n ! . (1.5)Note that lim λ −→ ∞ X n = k S ( x )2 ,λ ( n, k ) t n n ! = ( e t − k k ! e xt .From (1.4), we get ( t + x ) n,λ = n X k =0 S ( x )2 ,λ ( n, k ) ( t ) k , n >
0, (1.6)where ( t ) = 1, ( t ) n = t ( t −
1) ( t − · · · ( t − ( n − n ≥ e ( x + y ) λ ( t ) = ∞ X n =0 ( x + y ) n,λ t n n ! = ∞ X n =0 n X k =0 S ( x )2 ,λ ( n, k ) ( y ) k t n n ! .The degenerate Stirling numbers of the first kind are defined by ([3]-[10])1 k ! (log λ (1 + t )) k = ∞ X n = k S ,λ ( n, k ) t n n ! , k ≥
0. (1.7)Note here that lim λ −→ S ,λ ( n, l ) = S ( n, l ) where S ( n, l ) are the Stirling numbers of thefirst kind given by [5] (log (1 + t )) k k ! = ∞ X n = k S ( n, k ) t n n ! , k ≥
0. (1.8) n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 3
The degenerate Stirling numbers of the second kind are defined by ([3]-[10])( e λ ( t ) − k k ! = ∞ X n = k S ( n, k ) t n n ! , k ≥
0. (1.9)Observe that lim λ −→ S ,λ ( n, l ) = S ( n, l ) where S ( n, l ) are the Stirling numbers of thesecond kind given by [5] ( e t − k k ! = ∞ X n = k S ( n, k ) t n n ! , k ≥
0. (1.10)The degenerate Bernoulli polynomials of the second kind are given by ([6], [8]) t log λ (1 + t ) (1 + t ) x = ∞ X n =0 b n,λ ( x ) t n n ! . (1.11)Note that lim λ −→ b n,λ ( x ) = b n ( x ) where b n ( x ) are the Bernoulli polynomials of the secondkind given by [6] t log (1 + t ) (1 + t ) x = ∞ X n =0 b n ( x ) t n n ! . (1.12)2. Degenerate Poly-Frobenius-Genocchi Numbers and Polynomials
In this section, we introduce and investigate the modified polyexponential functions. Wegive some identities and explicit relations for the modified degenerate polyexponential func-tions. We define the degenerate poly-Frobenius-Genocchi polynomials. Also, we give somerelations and identities for these polynomials.In [2], Boyadzhiev introduced the polyexponential function, Kim et al. in ([6], [7]) con-sidered and investigated the polyexponential functions and the degenerate polyexponentialfunctions.The polyexponential functions are defined by ([3]-[11], [14])Ei k ( x ) = ∞ X n =1 x n n k ( n − k ∈ Z . (2.1)For k = 1, Ei ( x ) = e x − k,λ ( x ) = ∞ X n =1 (1) n,λ n k ( n − x n , λ ∈ R . (2.2)Note that Ei ,λ ( x ) = ∞ X n =1 (1) n,λ x n n ! = e λ ( x ) − Burak Kurt
For k ∈ Z and by means of the modified degenerate polyexponential functions. We definethe degenerate poly-Frobenius-Genocchi polynomials by the following generating functions. ∞ X n =0 F G ( k ) n,λ ( x, u ) t n n ! = (1 − u ) Ei k,λ (log λ (1 + t )) e λ ( t ) − u e xλ ( t ) . (2.3)When x = 0, F G ( k ) n,λ ( u ) := F G ( k ) n,λ (0 , u ) are called the degenerate poly-Frobenius-Genocchinumbers, where log λ ( t ) = λ (cid:0) t λ − (cid:1) is the compositional inverse of e λ ( t ) satisfyinglog λ ( e λ ( t )) = e λ (log λ (1 + t )) = t .For k = 1 and u = −
1, we get the degenerate Genocchi polynomials ∞ X n =0 F G (1) n,λ ( x, − t n n ! = 2 Ei ,λ (log λ (1 + t )) e λ ( t ) + 1 e xλ ( t )= 2 te λ ( t ) + 1 e xλ ( t ) = ∞ X n =0 G n,λ ( x ) t n n ! .From (2.3), we can write the following equations F G ( k ) n,λ ( x, u ) = n X m =0 (cid:18) nm (cid:19) F G ( k ) n,λ ( x ) n − m,λ , (i) F G ( k ) n,λ ( x + y, u ) = n X m =0 (cid:18) nm (cid:19) F G ( k ) n,λ ( x, u ) ( y ) n − m,λ , (ii)= n X m =0 (cid:18) nm (cid:19) F G ( k ) n,λ ( y, u ) ( x ) n − m,λ . F G ( k ) n,λ ( x + y, u ) = n X m =0 (cid:18) nm (cid:19) F G ( k ) n,λ ( x + y ) n − m,λ . (iii)By (1.8) and (2.2), we getEi k,λ (log λ (1 + t )) = ∞ X n =1 (1) n,λ (log λ (1 + t )) n n k ( n − ∞ X n =1 (1) n,λ ( n − k ∞ X m = n S ,λ ( m, n ) t n n != t ∞ X m =0 m +1 X n =1 (1) n,λ n k − S ,λ ( m + 1 , n ) m + 1 t m m ! . (2.4)Using (2.3) and (2.4), we get ∞ X n =0 F G ( k ) n,λ ( x, u ) t n n ! = (1 − u ) e xq ( t ) e λ ( t ) − u Ei k,λ (log λ (1 + t )) n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 5 = t ∞ X l =0 F G l,λ ( x, u ) t l l ! ∞ X m =0 m + 1 m +1 X j =1 (1) j,λ j k − S ,λ ( m + 1 , j ) t m m ! .By using Cauchy product and comparing the coefficients of t n n ! the above equations, wehave the following theorem. Theorem 1.
For n ≥ , we have F G ( k ) n,λ ( x, u ) = n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =1 (1) j,λ j k − S ,λ ( m + 1 , j ) F G n − − m,λ ( x, u ) . From (2.3), we write as ∞ X n =0 F G ( k ) n,λ ( x, u ) t n n ! ( e q ( t ) − u ) = (1 − u ) Ei k,λ (log λ (1 + t )) e xλ ( t )and = (1 − u ) ∞ X n =0 n X m =0 (cid:18) nm (cid:19) m +1 X j =1 (1) j,λ j k − S ,λ ( m + 1 , j ) m + 1 ( x ) n − m,λ ! t n +1 n ! . (2.5)Comparing the coefficients of both sides in (2.5).We have the following theorem. Theorem 2.
For n ≥ , we have n X m =0 (cid:18) nm (cid:19) F G ( k ) m,λ ( x, u ) (1) n − m,λ − u F G ( k ) n,λ ( x, u )= (1 − u ) n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =1 (1) j,λ j k − S ,λ ( m + 1 , j ) ( x ) n − − m,λ . From (2.2), we note that ddx Ei k,λ ( x ) = ddx ∞ X n =1 (1) j,λ ( n − n k x n = 1 x Ei k − ,λ ( x ) . (2.6)Thus, by (2.5), we get Ei k,λ ( x ) = x Z t Ei k − ,λ ( t ) dt = x Z t t Z · · · t t Z t | {z } ( k −
2) times Ei ,λ ( x ) dt · · · dt | {z } ( k −
2) times
Burak Kurt = x Z t t Z · · · t t Z t | {z } ( k −
2) times ( e λ ( t ) − dt · · · dt | {z } ( k −
2) times ,where k ∈ Z + with k ≥ k = 2 ∞ X n =0 F G (2) n,λ ( x, u ) t n n ! = 1 − ue λ ( t ) − u t Z t log λ (1 + t ) (1 + t ) λ − dt = 1 − ue λ ( t ) − u ∞ X m =0 b m,λ ( λ − m + 1 t m m ! = ∞ X l =0 H l,λ (0 , u ) t l l ! ∞ X m =0 b m,λ ( λ − m + 1 t m m ! .From the last equations, we have the following theorem. Theorem 3.
For n ≥ , we have F G (2) n,λ ( x, u ) = n X m =0 (cid:18) nm (cid:19) H n − m,λ (0 , u ) b m,λ ( λ − m + 1 ,where H n,λ (0 , u ) is degenerate Frobenius-Euler numbers. Recently, Masjed-Jamai et al in [13] and Srivastava et al in ([15], [16]) introduced a newtype parametric Euler numbers and polynomials as2 e t + 1 e pt cos ( qt ) = ∞ X n =0 E ( c ) n ( p, q ) t n n !and 2 e t + 1 e pt sin ( qt ) = ∞ X n =0 E ( s ) n ( p, q ) t n n ! ,where e pt cos ( qt ) = ∞ X n =0 C n ( p, q ) t n n !and e pt sin ( qt ) = ∞ X n =0 S n ( p, q ) t n n ! .3. Degenerate Poly-Frobenius-Genocchi Polynomials of Complex Variables
In this section, we define the Frobenius-Genocchi polynomials of the complex variables.We consider the degenerate cosine function and the degenerate sine function. Using thedegenerate cosine function and the degenerate sine function, we introduce the cosine degen-erate poly-Frobenius-Genocchi polynomials and the sine degenerate poly-Frobenius-Genocchipolynomials. n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 7
From (2.3), we write as ∞ X n =0 F G ( k ) n,λ ( x + iy ; u ) t n n ! = (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e ( x + iy ) λ ( t )= (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e xλ ( t ) h cos ( y ) λ ( t ) + i sin ( y ) λ ( t ) i (3.1)and ∞ X n =0 F G ( k ) n,λ ( x − iy ; u ) t n n != (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e xλ ( t ) h cos ( y ) λ ( t ) − i sin ( y ) λ ( t ) i . (3.2)By (3.1) and (3.2), we get(1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e xλ ( t ) cos ( y ) λ ( t )= ∞ X n =0 F G ( k ) n,λ ( x + iy ; u ) + F G ( k ) n,λ ( x − iy ; u )2 t n n ! (3.3)and (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e xλ ( t ) sin ( y ) λ ( t )= ∞ X n =0 F G ( k ) n,λ ( x + iy ; u ) − F G ( k ) n,λ ( x − iy ; u )2 i t n n ! . (3.4)Using (1.4), we define the degenerate cosine-functions and the degenerate sine-functionsas cos ( y ) λ ( t ) = e ( iy ) λ ( t ) + e ( − iy ) λ ( t )2 = cos (cid:16) yλ log (1 + λt ) (cid:17) (3.5)and sin ( y ) λ ( t ) = e ( iy ) λ ( t ) − e ( − iy ) λ ( t )2 i = sin (cid:16) yλ log (1 + λt ) (cid:17) , (3.6)where lim λ −→ cos ( y ) λ ( t ) = cos( yt ) and lim λ −→ sin ( y ) λ ( t ) = sin( yt ).Now, we define the cosine degenerate poly-Frobenius-Genocchi polynomials and the sinedegenerate poly-Frobenius-Genocchi polynomials, respectively; ∞ X n =0 F G [ k,c ] n,λ ( x, y ; u ) t n n ! = (1 − u ) Ei k,λ (log λ (1 + t )) e λ ( t ) − u e xλ ( t ) cos ( y ) λ ( t ) (3.7)and ∞ X n =0 F G [ k,s ] n,λ ( x, y ; u ) t n n ! = (1 − u ) Ei k,λ (log λ (1 + t )) e λ ( t ) − u e xλ ( t ) sin ( y ) λ ( t ) . (3.8) Burak Kurt
From (1.4), we write e ( iy ) λ ( t ) = ∞ X n =0 ( iy ) n,λ t n n ! and e ( − iy ) λ ( t ) = ∞ X n =0 ( − iy ) n,λ t n n ! .Using (3.5) and (3.6), we getcos ( y ) λ ( t ) = 12 ∞ X n =0 (cid:16) ( iy ) n,λ + ( − iy ) n,λ (cid:17) t n n ! (3.9)and sin ( y ) λ ( t ) = 12 i ∞ X n =0 (cid:16) ( iy ) n,λ − ( − iy ) n,λ (cid:17) t n n ! . (3.10)By (1.4), (3.9) and (1.4), (3.10), we have the following equations, respectively, e xλ ( t ) cos ( y ) λ ( t ) = 12 ∞ X n =0 n X k =0 (cid:18) nk (cid:19) ( x ) n − k,λ (cid:16) ( iy ) n,λ + ( − iy ) n,λ (cid:17) t n n ! (3.11)and e xλ ( t ) sin ( y ) λ ( t ) = 12 i ∞ X n =0 n X k =0 (cid:18) nk (cid:19) ( x ) n − k,λ (cid:16) ( iy ) n,λ − ( − iy ) n,λ (cid:17) t n n ! . (3.12)From (3.7) and (3.11), we write ∞ X n =0 F G [ k,c ] n,λ ( x, y ; u ) t n n ! = (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e xq ( t ) cos ( y ) λ ( t )= ∞ X n =0 F G ( k ) n,λ t n n ! 12 ∞ X n =0 n X k =0 (cid:18) nk (cid:19) ( x ) n − k,λ (cid:16) ( iy ) n,λ + ( − iy ) n,λ (cid:17) t n n ! .Using the Cauchy product and comparing the coefficients, we have F G [ k,c ] n,λ ( x, y ; u ) = 12 n X j =0 (cid:18) nj (cid:19) F G ( k ) n − j,λ j X k =0 (cid:18) jk (cid:19) ( x ) j − k,λ (cid:16) ( iy ) k,λ + ( − iy ) k,λ (cid:17) . (3.13)From (3.8) and (3.11), similarly, we have F G [ k,s ] n,λ ( x, y ; u ) = 12 i n X j =0 (cid:18) nj (cid:19) F G ( k ) n − j,λ j X k =0 (cid:18) jk (cid:19) ( x ) j − k,λ (cid:16) ( iy ) k,λ − ( − iy ) k,λ (cid:17) . (3.14)From (3.13) and (3.14), we have the following theorems. Theorem 4.
The following relations hold true:
F G [ k,c ] n,λ ( x, y ; u ) = 12 n X j =0 (cid:18) nj (cid:19) F G ( k ) n − j,λ j X k =0 (cid:18) jk (cid:19) ( x ) j − k,λ (cid:16) ( iy ) k,λ + ( − iy ) k,λ (cid:17) n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 9 and F G [ k,s ] n,λ ( x, y ; u ) = 12 i n X j =0 (cid:18) nj (cid:19) F G ( k ) n − j,λ j X k =0 (cid:18) jk (cid:19) ( x ) j − k,λ (cid:16) ( iy ) k,λ − ( − iy ) k,λ (cid:17) . Now, we define the degenerate two parametric C n,λ ( x, y ) and S n,λ ( x, y ) polynomials, re-spectively, e ( x ) λ ( t ) cos ( y ) λ ( t ) = ∞ X n =0 C n,λ ( x, y ) t n n ! (3.15)and e ( x ) λ ( t ) sin ( y ) λ ( t ) = ∞ X n =0 S n,λ ( x, y ) t n n ! . (3.16)From (1.4) and (3.9), we get C n,λ ( x, y ) = 12 n X k =0 (cid:18) nk (cid:19) ( x ) n − k,λ (cid:16) ( iy ) k,λ + ( − iy ) k,λ (cid:17) .Similarly, (1.4) and (3.10), we get S n,λ ( x, y ) = 12 i n X k =0 (cid:18) nk (cid:19) ( x ) n − k,λ (cid:16) ( iy ) k,λ − ( − iy ) k,λ (cid:17) .From (2.4), (3.7) and (3.11), we write( e q ( t ) − u ) ∞ X n =0 F G [ k,c ] n,λ ( x, y ; u ) t n n != (1 − u ) Ei k,λ (log λ (1 + t )) e ( x ) q ( t ) cos ( y ) λ ( t ) .The left hand side of this equation is ∞ X n =0 ( n X l =0 (cid:18) nl (cid:19) (1) n − l,λ F G [ k,c ] l,λ ( x, y ; u ) − u F G [ k,c ] n,λ ( x, y ; u ) ) t n n ! . (3.17)The right hand side of this equation is1 − u ∞ X n =0 n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =0 (1) j,λ j k − S ,λ ( m + 1 , j ) × n − − m X l =0 (cid:18) n − − ml (cid:19) ( x ) n − − m,λ (cid:16) ( iy ) n − − m,λ + ( − iy ) n − − m,λ (cid:17) t n n ! . (3.18)From (3.17) and (3.18), we get2 ( n X l =0 (cid:18) nl (cid:19) (1) n − l,λ F G [ k,c ] l,λ ( x, y ; u ) − u F G [ k,c ] n,λ ( x, y ; u ) ) Burak Kurt = (1 − u ) ( n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =0 (1) j,λ j k − S ,λ ( m + 1 , j ) × n − − m X l =0 (cid:18) n − − ml (cid:19) ( x ) n − − m,λ (cid:16) ( iy ) n − − m,λ + ( − iy ) n − − m,λ (cid:17)) . (3.19)Similarly, (2.4), (3.8) and (3.12)2 i ( n X l =0 (cid:18) nl (cid:19) (1) n − l,λ F G [ k,s ] l,λ ( x, y ; u ) − u F G [ k,s ] n,λ ( x, y ; u ) ) = (1 − u ) ( n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =0 (1) j,λ j k − S ,λ ( m + 1 , j ) × n − − m X l =0 (cid:18) n − − ml (cid:19) ( x ) n − − m,λ (cid:16) ( iy ) n − − m,λ − ( − iy ) n − − m,λ (cid:17)) . (3.20) Theorem 5.
The following relations hold true: ( n X l =0 (cid:18) nl (cid:19) (1) n − l,λ F G [ k,c ] l,λ ( x, y ; u ) − u F G [ k,c ] n,λ ( x, y ; u ) ) = (1 − u ) ( n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =0 (1) j,λ j k − S ,λ ( m + 1 , j ) × n − − m X l =0 (cid:18) n − − ml (cid:19) ( x ) n − − m,λ (cid:16) ( iy ) n − − m,λ + ( − iy ) n − − m,λ (cid:17)) and i ( n X l =0 (cid:18) nl (cid:19) (1) n − l,λ F G [ k,s ] l,λ ( x, y ; u ) − u F G [ k,s ] n,λ ( x, y ; u ) ) = (1 − u ) ( n n − X m =0 (cid:0) n − m (cid:1) m + 1 m +1 X j =0 (1) j,λ j k − S ,λ ( m + 1 , j ) × n − − m X l =0 (cid:18) n − − ml (cid:19) ( x ) n − − m,λ (cid:16) ( iy ) n − − m,λ − ( − iy ) n − − m,λ (cid:17)) . From (1.6) and (3.7), ∞ X n =0 F G [ k,c ] n,λ ( x + x , y ; u ) t n n ! = e ( x + x ) λ ( t ) (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u cos ( y ) λ ( t )= ∞ X m =0 m X k =0 S ( x )2 ,λ ( m, k ) ( x ) k t m m ! ∞ X l =0 F G [ k,c ] l,λ (0 , y ; u ) t l l ! n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 11 = ∞ X n =0 ( n X m =0 (cid:18) nm (cid:19) m X k =0 S ( x )2 ,λ ( m, k ) ( x ) k F G [ k,c ] n − m,λ (0 , y ; u ) ) t n n ! . (3.21)From (1.6) and (3.8), we get ∞ X n =0 F G [ k,s ] n,λ ( x + x , y ; u ) t n n ! = e ( x + x ) λ ( t ) (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u sin ( y ) λ ( t )= ∞ X n =0 ( n X m =0 (cid:18) nm (cid:19) m X k =0 S ( x )2 ,λ ( m, k ) ( x ) k F G [ k,s ] n − m,λ (0 , y ; u ) ) t n n ! . (3.22)Comparing the coefficients of t n n ! both sides the equations (3.21) and (3.22), we have thefollowing theorem. Theorem 6.
The following relations hold true:
F G [ k,c ] n,λ ( x + x , y ; u ) = n X m =0 (cid:18) nm (cid:19) m X k =0 S ( x )2 ,λ ( m, k ) ( x ) k F G [ k,c ] n − m,λ (0 , y ; u ) and F G [ k,s ] n,λ ( x + x , y ; u ) = n X m =0 (cid:18) nm (cid:19) m X k =0 S ( x )2 ,λ ( m, k ) ( x ) k F G [ k,s ] n − m,λ (0 , y ; u ) . Now, for our use in the present investigation, we find the expressions of cos ( x + x ) λ ( t ) andsin ( x + x ) λ ( t ).From (3.5), we get cos ( x + x ) λ ( t ) = cos (cid:18) x + x λ log (1 + λt ) (cid:19) = cos (cid:16) x λ log (1 + λt ) (cid:17) cos (cid:16) x λ log (1 + λt ) (cid:17) − sin (cid:16) x λ log (1 + λt ) (cid:17) sin (cid:16) x λ log (1 + λt ) (cid:17) = cos ( x ) λ ( t ) cos ( x ) λ ( t ) − sin ( x ) λ ( t ) sin ( x ) λ ( t ) . (3.23)Putting (3.23), x = x = x , we getcos (2 x ) λ ( t ) = (cid:16) cos ( x ) λ ( t ) (cid:17) − (cid:16) sin ( x ) λ ( t ) (cid:17) .By (3.6), we get sin ( x + x ) λ ( t ) = sin (cid:18) x + x λ log (1 + λt ) (cid:19) = sin ( x ) λ ( t ) cos ( x ) λ ( t ) + cos ( x ) λ ( t ) sin ( x ) λ ( t ) . (3.24)Setting (3.24), x = x = x , we getsin (2 x ) λ ( t ) = 2 cos ( x ) λ ( t ) sin ( x ) λ ( t ) . Burak Kurt
From (3.15) and (3.23), we write ∞ X n =0 C n,λ ( x + x , y + y ) t n n ! = e ( x + x ) λ ( t ) cos ( y + y ) λ ( t )= e ( x ) λ ( t ) e ( x ) λ ( t ) (cid:16) cos ( y ) λ ( t ) cos ( y ) λ ( t ) − sin ( y ) λ ( t ) sin ( y ) λ ( t ) (cid:17) = ∞ X n =0 C n,λ ( x , y ) t n n ! ∞ X n =0 C n,λ ( x , y ) t n n ! − ∞ X n =0 S n,λ ( x , y ) t n n ! ∞ X n =0 S n,λ ( x , y ) t n n ! . (3.25)Using (3.16) and (3.24), we write ∞ X n =0 S n,λ ( x + x , y + y ) t n n ! = e ( x + x ) λ ( t ) sin ( y + y ) λ ( t )= ∞ X n =0 S n,λ ( x , y ) t n n ! ∞ X n =0 C n,λ ( x , y ) t n n ! + ∞ X n =0 C n,λ ( x , y ) t n n ! ∞ X n =0 S n,λ ( x , y ) t n n ! . (3.26)By using Cauchy product above the equations (3.25) and (3.26), we have the followingtheorem. Theorem 7.
The following relations hold true: C n,λ ( x + x , y + y ) = n X k =0 (cid:18) nk (cid:19) { C n − k,λ ( x , y ) C k,λ ( x , y ) − S n − k,λ ( x , y ) S k,λ ( x , y ) } (3.27) and S n,λ ( x + x , y + y ) = n X k =0 (cid:18) nk (cid:19) { S n − k,λ ( x , y ) C k,λ ( x , y ) + C n − k,λ ( x , y ) S k,λ ( x , y ) } . (3.28)Setting x = x = x and y = y = y in (3.27) and (3.28), we have respectively, C n,λ (2 x, y ) = n X k =0 (cid:18) nk (cid:19) { C n − k,λ ( x, y ) C k,λ ( x, y ) − S n − k,λ ( x, y ) S k,λ ( x, y ) } and S n,λ (2 x, y ) = 2 n X k =0 (cid:18) nk (cid:19) S n − k,λ ( x, y ) C k,λ ( x, y ) .From (3.7) and (3.22), we write ∞ X n =0 F G [ k,c ] n,λ ( x + x , y + y ; u ) t n n != (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e ( x ) λ ( t ) e ( x ) λ ( t ) n cos ( x ) λ ( t ) cos ( x ) λ ( t ) − sin ( y ) λ ( t ) sin ( y ) λ ( t ) o n the degenerate poly-Frobenius-Genocchi polynomials of complex variables 13 = ∞ X m =0 F G [ k,c ] m,λ ( x , y ; u ) t m m ! ∞ X k =0 C k,λ ( x , y ) t k k ! − ∞ X m =0 F G [ k,s ] m,λ ( x , y ; u ) t m m ! ∞ X k =0 S k,λ ( x , y ) t k k ! . (3.29)Using (3.8) and (3.24), we write ∞ X n =0 F G [ k,s ] n,λ ( x + x , y + y ; u ) t n n != (1 − u ) Ei k,λ (log λ (1 + t )) e q ( t ) − u e ( x ) λ ( t ) e ( x ) λ ( t ) n sin ( x ) λ ( t ) cos ( x ) λ ( t ) + cos ( y ) λ ( t ) sin ( y ) λ ( t ) o = ∞ X m =0 F G [ k,s ] m,λ ( x , y ; u ) t m m ! ∞ X k =0 C k,λ ( x , y ) t k k !+ ∞ X m =0 F G [ k,s ] m,λ ( x , y ; u ) t m m ! ∞ X k =0 S k,λ ( x , y ) t k k ! . (3.30)Using Cauchy product (3.29) and (3.30), we have the following theorem. Theorem 8.
The following relations hold true:
F G [ k,c ] n,λ ( x + x , y + y ; u )= n X k =0 (cid:18) nk (cid:19) n F G [ k,c ] n − k,λ ( x , y ; u ) C k,λ ( x , y ) − F G [ k,s ] n − k,λ ( x , y ; u ) S k,λ ( x , y ) o (3.31) and F G [ k,s ] n,λ ( x + x , y + y ; u )= n X k =0 (cid:18) nk (cid:19) n F G [ k,s ] n − k,λ ( x , y ; u ) C k,λ ( x , y ) + F G [ k,c ] n − k,λ ( x , y ; u ) S k,λ ( x , y ) o . (3.32)Putting x = x = x and y = y = y in (3.31) and (3.32), respectively, we have F G [ k,c ] n,λ (2 x, y ; u )= n X k =0 (cid:18) nk (cid:19) n F G [ k,c ] n − k,λ ( x, y ; u ) C k,λ ( x, y ) − F G [ k,s ] n − k,λ ( x, y ; u ) S k,λ ( x, y ) o and F G [ k,s ] n,λ (2 x, y ; u )= n X k =0 (cid:18) nk (cid:19) n F G [ k,s ] n − k,λ ( x, y ; u ) C k,λ ( x, y ) + F G [ k,c ] n − k,λ ( x, y ; u ) S k,λ ( x, y ) o . Burak Kurt Conclusion
Kim and Kim [7] considered the polyexponential and unipoly functions. Kim et al. ([3] and[11]) defined and investigated the new type modified degenerate polyexponential function,the degenerate poly-Bernoulli polynomials and the degenerate poly-Genocchi polynomials.Motivated by these studying, we introduce the degenerate poly-Frobenius-Genocchi polyno-mials of the complex variables. We also give their some interesting properties and identities.As one of our future projects, we would like to continue to do researcher on degenerateversions of various special numbers and polynomials.
Acknowledgement 1.
The present investigation was supported, in part, by the ScientificResearch Project Administration of the University of Akdeniz.
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