Extended fermionic p -adic integrals on Z p
aa r X i v : . [ m a t h . N T ] D ec EXTENDED FERMIONIC p -ADIC INTEGRALS ON Z p FENG QI, SERKAN ARACI, AND MEHMET ACIKGOZ
Abstract.
In the paper, using the extended fermionic p -adic integral on Z p ,the authors find some applications of the umbral calculus. From these appli-cations, the authors derive some identities on the weighted Euler numbers andpolynomials. In other words, the authors investigate systematically the classof Sheffer sequences in connection with the generating function of the weightedEuler polynomials. Preliminaries
Let C denote the set of complex numbers, F the set of all formal power series in t over C with(1.1) F = ( f ( t ) = ∞ X k =0 a k t k k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k ∈ C ) , P = C [ x ], P ∗ the vector space of all linear functionals on P , and h L | p ( x ) i the actionof the linear functional L on the polynomial p ( x ).It is well-known that the vector space operation on P ∗ is defined by h L + M | p ( x ) i = h L | p ( x ) i + h M | p ( x ) i and h cL | p ( x ) i = c h L | p ( x ) i , (1.2)where c is a complex constant.The formal power series is defined by(1.3) f ( t ) = ∞ X k =0 a k t k k ! ∈ F which describes a linear functional on P as h f ( t ) | x n i = a n for all n ≥
0. Inparticular,(1.4) (cid:10) t k | x n (cid:11) = n ! δ n,k , where δ n,k stands for the Kronecker delta. If we take(1.5) f L ( t ) = ∞ X k =0 (cid:10) L | x k (cid:11) t k k ! , then(1.6) h f L ( t ) | x n i = h L | x n i . Additionally, the map L → f L ( t ) is a vector space isomorphism from P ∗ onto F .Henceforth, F will denote both the algebra of the formal power series in t and thevector space of all linear functionals on P . So an element f ( t ) of F will be thought Mathematics Subject Classification.
Primary 11B68; Secondary 11S80.
Key words and phrases.
Appell sequence, Sheffer sequence, Euler number, Euler polynomial,formal power series, fermionic p -adic integral on Z p , umbral calculus. of as both a formal power series and a linear functional and F will be called anumbral algebra.It is well-known that h e yt | x n i = y n , which implies that(1.7) h e yt | p ( x ) i = p ( y ) . We note that for all f ( t ) in F (1.8) f ( t ) = ∞ X k =0 (cid:10) f ( t ) | x k (cid:11) t k k !and for all polynomials p ( x )(1.9) p ( x ) = ∞ X k =0 (cid:10) t k | p ( x ) (cid:11) x k k ! . The order o ( f ( t )) of the power series f ( t ) = 0 is the smallest integer k for which a k does not vanish. We say that o ( f ( t )) = ∞ if f ( t ) = 0. It is clear that o ( f ( t ) g ( t )) = o ( f ( t )) + o ( g ( t )) and o ( f ( t ) + g ( t )) ≥ min { o ( f ( t )) , o ( g ( t )) } . A series f ( t ) has a multiplicative inverse, denoted by f ( t ) − or f ( t ) , if o ( f ( t )) = 0.Such a series is called an invertible series. A series f ( t ) satisfying o ( f ( t )) = 1 iscalled a delta series. For f ( t ) , g ( t ) ∈ F , we have h f ( t ) g ( t ) | p ( x ) i = h f ( t ) | g ( t ) p ( x ) i . A delta series f ( t ) has a compositional inverse ¯ f ( t ) such that f (cid:0) ¯ f ( t ) (cid:1) = ¯ f ( f ( t )) = t. By (1.8), it follows that(1.10) p ( k ) ( x ) = d k p ( x )d x k = ∞ X ℓ = k (cid:10) t ℓ | p ( x ) (cid:11) ℓ ! k − Y i =0 ( ℓ − i ) x ℓ − k and(1.11) p ( k ) (0) = (cid:10) t k | p ( x ) (cid:11) = (cid:10) | p ( k ) ( x ) (cid:11) . The relation (1.10) implies that(1.12) t k p ( x ) = p ( k ) ( x ) = d k p ( x )d x k and(1.13) e yt p ( x ) = p ( x + y ) . Let S n ( x ) denote a polynomial with degree n . Let f ( t ) be a delta series and g ( t )an invertible series. Then there exists a unique sequence S n ( x ) such that (cid:10) g ( t ) f k ( t ) | S n ( x ) (cid:11) = n ! δ n,k for all n, k ≥
0. Such a sequence S n ( x ) is called a Sheffer sequence for ( g ( t ) , f ( t ))or say that S n ( t ) is Sheffer for ( g ( t ) , f ( t )).The Sheffer sequence for (1 , f ( t )) is called an associated sequence for f ( t ) orsay that S n ( x ) is associated to f ( t ). The Sheffer sequence for ( g ( t ) , t ) is called anAppell sequence for g ( t ) or say that S n ( x ) is Appell for g ( t ).Let p ( x ) ∈ P . Then h f ( t ) | xp ( x ) i = h ∂ t f ( t ) | p ( x ) i = h f ′ ( t ) | p ( x ) i XTENDED FERMIONIC p -ADIC INTEGRALS ON Z p and h e ty + 1 | p ( x ) i = p ( y ) + p (0) . Let S n ( x ) be Sheffer for ( g ( t ) , f ( t )). Then h ( t ) = ∞ X k =0 h h ( t ) | S k ( x ) i k ! g ( t ) f k ( t ) , h ( t ) ∈ F , (1.14) p ( x ) = ∞ X k =0 h g ( t ) f k ( t ) | p ( x ) i k ! S k ( x ) , p ( x ) ∈ P , (1.15) e y ¯ f ( t ) g (cid:0) ¯ f ( t ) (cid:1) = ∞ X k =0 S k ( y ) t k k ! , y ∈ C , (1.16) f ( t ) S n ( x ) = nS n − ( x ) . (1.17)Moreover, we have(1.18) h f ( t ) f ( t ) · · · f m ( t ) | x n i = X (cid:18) ni , . . . , i m (cid:19) m Y j =1 (cid:10) f j ( t ) | x i j (cid:11) , where f ( t ) , f ( t ) , . . . , f m ( t ) ∈ F and the sum is taken over all nonnegative integers i , . . . , i m such that i + · · · + i m = n .For details on the above knowledge, please refer to [8, 9, 22, 23, 24, 25, 26] andplenty of references therein.Let p be a fixed odd prime number. In what follows, we use Z p to denote the ringof p -adic rational integers, Q the field of rational numbers, Q p the field of p -adicrational numbers, and C p the completion of algebraic closure of Q p . Let N be theset of natural numbers and N ∗ = { } ∪ N . The p -adic absolute value is defined by | p | p = p − . We also assume that | q − | p < U D ( Z p )be the space of uniformly differentiable functions on Z p . For f ∈ U D ( Z p ), thefermionic p -adic integral on Z p is defined by Kim (see [1, 2, 3, 4]) as(1.19) I − ( f ) = Z Z p f ( a ) d µ − ( a ) = lim n →∞ p n − X a =0 f ( a )( − a . Hence, we have(1.20) I − ( f ) + I − ( f ) = 2 f (0) , where f ( a ) = f ( a + 1). For detailed information on these notions, see [5, 7, 11, 12,13, 14, 15, 16, 17, 18, 19].Now let us consider Kim’s p -adic fermionic integral on Z p . For | − w | p < I w − ( f ) = Z Z p w a f ( a ) d µ − ( a ) = lim n →∞ p n − X a =0 w a f ( a )( − a , where I w − ( f ) is the extended fermionic p -adic integral on Z p . Letting f ( x ) = e t ( x + a ) ∈ U D ( Z p ) in this equation yields(1.22) Z Z p w a e t ( x + a ) d µ − ( a ) = 2 we t + 1 e tx = ∞ X n =0 E n,w ( x ) t n n ! , F. QI, S. ARACI, AND M. ACIKGOZ where E n,w ( x ) is the weighted Euler polynomials defined in [20]. Specially, thequantity E n,w (0) = E n,w is the weighted Euler numbers. The relation betweenweighted Euler numbers and weighted Euler polynomials is given by(1.23) E n,w ( x ) = n X ℓ =0 (cid:18) nℓ (cid:19) x ℓ E n − ℓ,w = ( x + E w ) n , with the usual convention of replacing ( E w ) n by E n,w . Combing this with (1.22)leads to(1.24) E n,w = Z Z p w a a n d µ − ( a ) and E n,w ( x ) = Z Z p w a ( x + a ) n d µ − ( a ) . In [9, 10], the authors studied applications of the umbral algebra to specialfunctions. In [21], the author gave some new interesting links to works of manymathematicians in the analytic number theory and the modern classical umbralcalculus. In [22, 23, 24], the authors established some properties of the umbral cal-culus for Frobenius-Euler polynomials, Euler polynomials, and other special func-tions. In [21], the authors investigated some new applications of the umbral calculusassociated with p -adic invariants integral on Z p .In this paper, by the same motivation as in [21] and using the extended fermionic p -adic integral on Z p , we will give some applications of the umbral calculus and,from these applications, derive some identities concerning weighted Euler numbers,weighted Euler polynomials, and weighted Euler polynomials of order k .2. On the extended fermionic p -adic integral on Z p Now we start out to state and prove our main results.
Theorem 2.1. If n ≥ , then E n,w ( x ) is an Appell sequence for g ( t ) = we t +12 .Proof. Suppose that S n ( x ) is an Appell sequence for g ( t ). Then, by (1.16), we have(2.1) 1 g ( t ) x n = S n ( x ) if and only if x n = g ( t ) S n ( x )for n ≥
0. Let g ( t ) = we t + 12 ∈ F . It is clear that g ( t ) is an invertible series. By (2.1), we have(2.2) ∞ X n =0 E n,w ( x ) t n n ! = 1 g ( t ) e xt . This means that(2.3) 1 g ( t ) x n = E n,w ( x ) . Making use of (1.16) gives(2.4) tE n,w ( x ) = E ′ n,w ( x ) = nE n − ,w ( x ) . Combining (2.3) and (2.4) results in Theorem 2.1. (cid:3)
Theorem 2.2.
Let g ( t ) = we t +12 ∈ F . Then for n ≥ E n +1 ,w ( x ) = (cid:20) x − g ′ ( t ) g ( t ) (cid:21) E n,w ( x ) . XTENDED FERMIONIC p -ADIC INTEGRALS ON Z p Proof.
By (1.24), we derive that ∞ X n =1 E n,w ( x ) t n n ! = xg ( t ) e xt − g ′ ( t ) e xt g ( t ) = ∞ X n =0 (cid:20) x g ( t ) x n − g ′ ( t ) g ( t ) 1 g ( t ) x n (cid:21) t n n ! . Considering (2.3) and the above equality, we discover E n +1 ,w ( x ) = xE n,w ( x ) − g ′ ( t ) g ( t ) E n,w ( x ) . Theorem 2.2 is thus proved. (cid:3)
Theorem 2.3.
For n ≥ , (2.6) E n +1 ,w ( x ) = (cid:20) x − g ′ ( t ) g ( t ) (cid:21) E n,w ( x ) , where g ′ ( t ) = d g ( t )d t .Proof. From (1.24), it is easy to see that ∞ X n =0 [ wE n,w ( x + 1) + E n,w ( x )] t n n ! = ∞ X n =0 (2 x n ) t n n ! . Comparing the coefficients on the both sides, we find(2.7) wE n,w ( x + 1) + E n,w ( x ) = 2 x n . From Theorem 2.2, it follows that(2.8) g ( t ) E n +1 ,w ( x ) = g ( t ) xE n,w ( x ) − g ′ ( t ) E n,w ( x )and ( we t + 1) E n +1 ,w ( x ) = ( we t + 1) xE n,w ( x ) − we t E n,w ( x ) . Consequently, we have wE n +1 ,w ( x + 1) + E n +1 ,w ( x )= w ( x + 1) E n,w ( x + 1) + xE n,w ( x ) − wE n,w ( x + 1) . Combining this with (2.7) and (2.8), we acquire the required conclusions. (cid:3)
Corollary 2.3.1.
For n ≥ , we have wE n +1 ( x + 1) + E n +1 ,w ( x ) = 2 x n +1 . Theorem 2.4.
For n ≥ , we have h f ( t ) | p ( x ) i = Z Z p w a p ( a ) d µ − ( a ) , (2.9) (cid:28) we t + 1 (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) (cid:29) = Z Z p w a p ( a ) d µ − ( a ) , (2.10) E n,w = (cid:28)Z Z p w a e at d µ − ( a ) (cid:12)(cid:12)(cid:12)(cid:12) x n (cid:29) . (2.11) F. QI, S. ARACI, AND M. ACIKGOZ
Proof.
Let us consider the linear functional f ( t ) satisfying(2.12) h f ( t ) | p ( x ) i = Z Z p w a p ( a ) d µ − ( a )for all polynomials p ( x ). Then we readily see that f ( t ) = ∞ X n =0 h f ( t ) | x n i n ! t n = ∞ X n =1 (cid:20)Z Z p w a a n d µ − ( a ) (cid:21) t n n ! = Z Z p w a e at d µ − ( a ) . Thus, we have(2.13) f ( t ) = Z Z p w a e at d µ − ( a ) = 2 we t + 1 . Therefore, by (2.12) and (2.13), we arrive at the theorem. (cid:3)
Theorem 2.5.
For p ( x ) ∈ P , we have (2.14) Z Z p w a p ( x + a ) d µ − ( a ) = Z Z p w a e at d µ − ( a ) p ( x ) = 2 we t + 1 p ( x ) . Equivalently, (2.15) E n,w ( x ) = Z Z p w a e at d µ − ( a ) x n = 2 we t + 1 x n . Proof.
From (1.24) and (2.11), we see that ∞ X n =0 (cid:20)Z Z p w a ( x + a ) n d µ − ( a ) (cid:21) t n n ! = Z Z p w a e ( x + a ) t d µ − ( a )= ∞ X n =0 (cid:20)Z Z p w a e at d µ − ( a ) x n (cid:21) t n n ! . (2.16)By this equality and (1.24), we see that for n ∈ N ∗ (2.17) E n,w ( x ) = Z Z p ( x + a ) n d µ − ( a ) = Z Z p w a e at d µ − ( a ) x n . As a result, we obtain the theorem. (cid:3)
Theorem 2.6.
For p ( x ) ∈ P and k ∈ N , we have (2.18) Z Z p · · · Z Z p | {z } k -times w a + ··· + a k p ( a + · · · + a k + x ) k Y j =1 d µ − ( a j )= (cid:18) we t + 1 (cid:19) k p ( x ) . In particular, E ( k ) n,w ( x ) = (cid:18) we t + 1 (cid:19) k x n = x n Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j ) . XTENDED FERMIONIC p -ADIC INTEGRALS ON Z p Consequently, E ( k ) n,w ( x ) ∼ (cid:18)(cid:18) we t + 12 (cid:19) k , t (cid:19) . Proof.
For | − w | p <
1, we consider the weighted Euler polynomials of order k .(2.19) Z Z p · · · Z Z p | {z } k -times w a + ··· + a k e ( a + ··· + a k + x ) t k Y j =1 d µ − ( a j )= (cid:18) we t + 1 (cid:19) k e xt = ∞ X n =0 E ( k ) n,w ( x ) t n n ! . where E ( k ) n,w (0) = E ( k ) n,w are the weighted Euler numbers of order k . Accordingly,(2.20) Z Z p · · · Z Z p | {z } k -times w a + ··· + a k ( a + · · · + a k ) n k Y j =1 d µ − ( a j )= X i + ··· + i k = n (cid:18) ni , . . . , i m (cid:19) k Y j =1 Z Z p w a j a i j j d µ − ( a j )= X i + ··· + i k = n (cid:18) ni , . . . , i m (cid:19) k Y j =1 E i j ,w = E ( k ) n,w . Thanks to (2.19) and (2.20), we have(2.21) E ( k ) n,w ( x ) = n X ℓ =0 (cid:18) nℓ (cid:19) x ℓ E ( k ) n − ℓ,w . From (2.20) and (2.21), we notice that E ( k ) n,w ( x ) is a monic polynomial of degree n with coefficients in Q . For k ∈ N , let us assume that(2.22) g ( k ) ( t ) = "Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j ) − = (cid:18) we t + 12 (cid:19) k . From this, we see that g ( k ) ( t ) is an invertible series. Due to (2.19) and (2.22), wereadily derive that1 g ( k ) ( t ) e xt = Z Z p · · · Z Z p | {z } k -times w a + ··· + a k e ( a + ··· + a k + x ) t k Y j =1 d µ − ( a j )= ∞ X n =0 E ( k ) n,w ( x ) t n n ! . Taking account of this and(2.23) tE ( k ) n,w ( x ) = nE ( k ) n − ,w ( x ) F. QI, S. ARACI, AND M. ACIKGOZ yields that E ( k ) n,w ( x ) is an Appell sequence for g ( k ) ( t ). Theorem 2.6 is proved. (cid:3) Theorem 2.7.
For p ( x ) ∈ P , we have (2.24) (cid:28)Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j ) (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) (cid:29) = Z Z p · · · Z Z p w a + ··· + a k p ( a + · · · + a k ) k Y j =1 d µ − ( a j ) . Furthermore, (cid:28)(cid:18) we t + 1 (cid:19) k (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) (cid:29) = Z Z p · · · Z Z p w a + ··· + a k p ( a + · · · + a k ) k Y j =1 d µ − ( a j ) , equivalently, E ( k ) n,w = (cid:28)Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j ) (cid:12)(cid:12)(cid:12)(cid:12) x n (cid:29) . Proof.
Let us take the linear functional f ( k ) ( t ) fulfilling(2.25) (cid:10) f ( k ) ( t ) | p ( x ) (cid:11) = Z Z p · · · Z Z p w a + ··· + a k p ( a + · · · + a k ) k Y j =1 d µ − ( a j )for all polynomials p ( x ). Then f ( k ) ( t ) = ∞ X n =0 (cid:10) f ( k ) ( t ) | x n (cid:11) n ! t n = ∞ X n =0 "Z Z p · · · Z Z p w a + ··· + a k ( a + · · · + a k ) n k Y j =1 d µ − ( a j ) t n n != Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j )= (cid:18) we t + 1 (cid:19) k . Therefore, we procure Theorem 2.7. (cid:3)
Remark . From (1.18), we notice that (cid:28)Z Z p · · · Z Z p w a + ··· + a k e ( a + ··· + a k ) t k Y j =1 d µ − ( a j ) (cid:12)(cid:12)(cid:12)(cid:12) x n (cid:29) = X i + ··· + i k = n (cid:18) ni , . . . , i k (cid:19) k Y ℓ =1 (cid:28)Z Z p w a ℓ e a ℓ t d µ − ( a ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) x i ℓ (cid:29) . Therefore, we have E ( k ) n,w = X i + ··· + i k = n (cid:18) ni , . . . , i k (cid:19) E i ,w · · · E i k ,w . XTENDED FERMIONIC p -ADIC INTEGRALS ON Z p Remark . Our applications to the weighted Euler polynomials, the weighted Eu-ler numbers, and the weighted Euler polynomials of order k seem to be interesting,because evaluating at w = 1 leads to Euler polynomials and Euler polynomials oforder k defined respectively by ∞ X n =0 E n ( x ) t n n ! = 2 e t + 1 e xt and ∞ X n =0 E ( k ) n ( x ) t n n ! = (cid:18) e t + 1 (cid:19) k e xt . It is also well known that E n ( x ) = Z Z p ( x + a ) n d µ − ( a )and E ( k ) n ( x ) = Z Z p · · · Z Z p ( a + · · · + a k + x ) n k Y j =1 d µ − ( a j ) . See [5, 6, 11, 13, 16] and related references therein.
References [1] S. Araci, M. Acikgoz, and F. Qi,
On the q -Genocchi numbers and polynomials with weightzero and their applications , Nonlinear Funct. Anal. Appl. (2013), no. 2, 193–203. 3[2] S. Araci, M. Acikgoz, and F. Qi, On the q -Genocchi numbers and polynomials with weightzero and their applications , available online at http://arxiv.org/abs/1202.2643 . 3[3] S. Araci, M. Acikgoz, F. Qi, and H. Jolany, A note on the modified q -Genocchi numbers andpolynomials with weight ( α, β ) and their interpolation function at negative integers , Fasc.Math. (2013), in press. 3[4] S. Araci, M. Acikgoz, F. Qi, and H. Jolany, A note on the modified q -Genocchi numbers andpolynomials with weight ( α, β ) and their interpolation function at negative integers , availableonline at http://arxiv.org/abs/1112.5902 . 3[5] S. Araci, M. Acikgoz, and J. J. Seo, Explicit formulas involving q -Euler numbers and poly-nomials , Abstr. Appl. Anal. (2012), Article ID 298531, 11 pages; Available online at http://dx.doi.org/10.1155/2012/298531 . 3, 9[6] S. Araci and D. Erdal, Higher order Genocchi, Euler polynomials associated with q -Bernsteintype polynomials , Honam Math. J. (2011), no. 2, 173–179. 9[7] S. Araci, D. Erdal, and J. J. Seo, A study on the fermionic p -adic q -integral representationon Z p associated with weighted q -Bernstein and q -Genocchi polynomials , Abstr. Appl. Anal. (2011), Article ID 649248, 10 pages; Available online at http://dx.doi.org/10.1155/2011/649248 . 3[8] S. Araci, X.-X. Kong, M. Acikgoz, and E. S¸en, A new approach to multivariate q -Eulerpolynomials by using umbral calculus , available online at http://arxiv.org/abs/1211.4062 .3[9] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials , Adv.Stud. Contemp. Math. (2012), no. 3, 433–438. 3, 4[10] R. Dere and Y. Simsek, Genocchi polynomials associated with the umbral algebra , Appl.Math. Comp. (2011), no. 3, 756–761; Available online at http://dx.doi.org/10.1016/j.amc.2011.01.078 . 4[11] M. Acikgoz and Y. Simsek,
On multiple interpolation function of the N¨orlund-type q -Eulerpolynomials , Abstr. Appl. Anal. (2009), Article ID 382574, 14 pages; Available onlineat http://dx.doi.org/10.1155/2009/382574 . 3, 9[12] T. Kim, Non-Archimedean q -integrals associated with multiple Changhee q -Bernoulli polyno-mials , Russ. J. Math. Phys. (2003), 91–98. 3[13] T. Kim, On p -adic interpolating function for q -Euler numbers and its derivatives , J. Math.Anal. Appl. (2008), 598–608; Available online at http://dx.doi.org/10.1016/j.jmaa.2007.07.027 . 3, 9[14] T. Kim, On the q -extension of Euler and Genocchi numbers , J. Math. Anal. Appl. (2007), 1458–1465; Available online at http://dx.doi.org/10.1016/j.jmaa.2006.03.037 . 3 [15] T. Kim, p -adic q -integrals associated with the Changhee-Barnes’ q -Bernoulli polynomials ,Integral Transforms Spec. Funct. (2004), 415–420; Available online at http://dx.doi.org/10.1080/10652460410001672960 . 3[16] T. Kim, Symmetry p -adic invariant integral on Z p for Bernoulli and Euler polynomials , J.Difference Equ. Appl. (2008), no. 12, 1267–1277; Available online at http://dx.doi.org/10.1080/10236190801943220 . 3, 9[17] T. Kim, The modified q -Euler numbers and polynomials , Adv. Stud. Contemp. Math. (2008), 161–170. 3[18] T. Kim, On a q -analogue of the p -adic log gamma functions and related integrals , J. NumberTheory (1999), 320–329; Available online at http://dx.doi.org/10.1006/jnth.1999.2373 . 3[19] T. Kim, Symmetry of power sum polynomials and multivariate fermionic p -adic invariantintegral on Z p , Russ. J. Math. Phys. (2009), no. 1, 93–96. 3[20] T. Kim, J. Choi, and H.-M. Kim, A note on the weighted Lebesgue Radon-Nikodym theoremwith respect to p -adic invariant integral on Z p , J. Appl. Math. Inform. (2012), no. 1-2,211–217. 4[21] D. S. Kim and T. Kim, Applications of umbral calculus associated with p -adic invariantsintegral on Z p , Abstr. Appl. Anal. (2012), Article ID 865721, 12 pages; Availableonline at http://dx.doi.org/10.1155/2012/865721 . 4[22] D. S. Kim and T. Kim, Some identities of Frobenius-Euler polynomials arising from umbralcalculus , Adv. Difference Equ. 2012, :196; Available online at http://dx.doi.org/10.1186/1687-1847-2012-196 . 3, 4[23] D. S. Kim and T. Kim,
Umbral calculus and Euler polynomials , available online at http://arxiv.org/abs/1211.6639 . 3, 4[24] D. S. Kim, T. Kim, and S. H. Rim,
Some identities of polynomials arising from umbralcalculus , available online at http://arxiv.org/abs/1211.3738 . 3, 4[25] M. Maldonado, J. Prada, and M. J. Senosiain,
Appell bases on sequence spaces , J. NonlinearMath. Phys. (2011), Suppl. 1, 189–194; Available online at http://dx.doi.org/10.1142/S1402925111001362 . 3[26] S. Roman, The Umbral Calculus , Dover Publ. Inc. New York, 2005. 3(Qi)
Department of Mathematics, College of Science, Tianjin Polytechnic Univer-sity, Tianjin City, 300387, China
E-mail address : [email protected], [email protected], [email protected] URL : http://qifeng618.wordpress.com (Araci) Department of Mathematics, Faculty of Science and Arts, University ofGaziantep, 27310 Gaziantep, Turkey
E-mail address : [email protected] (Acikgoz) Department of Mathematics, Faculty of Science and Arts, University ofGaziantep, 27310 Gaziantep, Turkey
E-mail address ::