A note on Fermionic p-adic integrals on Zp and umbral calculus
aa r X i v : . [ m a t h . N T ] N ov A NOTE ON FERMIONIC p -ADIC INTEGRALS ON Z p ANDUMBRAL CALCULUS
TAEKYUN KIM, DAE SAN KIM, SANGTAE JEONG AND SEOG-HOON RIM
Abstract.
In this paper we study some properties of the fermionic p -adicintegrals on Z p arising from the umbral calculus. Introduction
Let p be a fixed odd prime number. Throughout this paper Z p , Q p and C p denote the ring of p -adic rational integers, the field of p -adic rational numbersand the completion of the algebraic closure of Q p , respectively. Let N be the set ofnatural numbers and Z + = N ∪{ } . Let C ( Z p ) be the space of continuous functionson Z p . For f ∈ C ( Z p ), the fermionic p -adic integral on Z p is defined by Z Z p f ( x ) dµ − ( x ) = lim N →∞ p N − X x =0 f ( x ) µ − ( x + p N Z p )= lim N →∞ p N − X x =0 f ( x )( − x , (see [1,2,11]) . (1)For n ∈ N , we have Z Z p f ( x + n ) dµ − ( x ) + ( − n − Z Z p f ( x ) dµ − ( x ) = 2 n − X l =0 ( − n − − l f ( l ) . (2)In the special case, n = 1, we note that Z Z p f ( x + 1) dµ − ( x ) + Z Z p f ( x ) dµ − ( x ) = 2 f (0) , (see [ 11]) . (3)Let F be the set of all formal power series in the variable t over C p with F = n f ( t ) = ∞ X k =0 a k k ! t k (cid:12)(cid:12)(cid:12) a k ∈ C p o . Let P = C p [ x ] and let P ∗ denote the vector space of all linear functionals on P . Theformal power series f ( t ) = ∞ X k =0 a k k ! t k ∈ F . (4)defines a linear functional on P by setting h f ( t ) | x n i = a n for all n ≥ , (see [ 7,14]) . (5) Thus, by (4) and (5), we get h t k | x n i = n ! δ n,k , ( n, k ≥ , (6)where δ n,k is the Kronecker symbol (see [7,14]). Here, F denotes both the algebraof formal power series in t and the vector space of all linear functionals on P , and soan element f ( t ) of F will be thought of as both a formal power series and a linearfunctional. We shall call F the umbral algebra. The umbral calculus is the studyof umbral algebra (see [7,14]).The order O ( f ( t )) of power series f ( t )( = 0) is the smallest integer k for which a k does not vanish (see [7,4]). The series f ( t ) has a multiplicative inverse, denotedby f ( t ) − or f ( t ) , if and only if O ( f ( t )) = 0. Such series is called an invertibleseries. A series f ( t ) for which O ( f ( t )) = 1 is called a delta series (see [7,14]). For f ( t ) , g ( t ) ∈ F , we have h f ( t ) g ( t ) | p ( x ) i = h f ( t ) | g ( t ) p ( x ) i = h g ( t ) | f ( t ) p ( x ) i . By (6),we get h e yt | x n i = y n , h e yt | p ( x ) i = p ( y ) , (see [ 7,14]) . (7)Let f ( t ) ∈ F . Then we note that f ( t ) = ∞ X k =0 h f ( t ) | x k i k ! t k , (8)and p ( x ) = ∞ X k =0 h t k | p ( x ) i k ! x k , for p ( x ) ∈ P , (see [14]) . (9)Let f ( t ) , f ( t ) , · · · , f m ( t ) ∈ F . It is known in [7,14] that h f ( t ) · · · f m ( t ) | x n i = X (cid:18) ni , · · · , i m (cid:19) h f ( t ) | x i i · · · h f m ( t ) | x i m i , (10)where the sum is over all nonnegative integers i , · · · , i m such that i + i + · · · + i m = n (see [7,14]).By (9), we get p ( k ) ( x ) = d k p ( x ) dx k = ∞ X l = k h t l | p ( x ) i l ! l ( l − · · · ( l − k + 1) x l − k = ∞ X l = k h t l | p ( x ) i (cid:18) lk (cid:19) k ! l ! x l − k . (11)Thus, from (11), we have p ( k ) (0) = h t k | p ( x ) i = h | p ( k ) ( x ) i , (12)and t k p ( x ) = p ( k ) ( x ) = d k p ( x ) dx k (see [7,14]) . (13)From (13), we note that e yt p ( x ) = p ( x + y ) (see [7,14]) . (14)In this paper, s n ( x ) denotes a polynomial of degree n . Let us assume that f ( t ) , g ( t ) ∈F with o ( f ( t )) = 1 and o ( g ( t )) = 1. Then there exists a unique sequence s n ( x ) NOTE ON FERMIONIC p -ADIC INTEGRALS ON Z p AND UMBRAL CALCULUS 3 of polynomials satisfying h g ( t ) f ( t ) k | s n ( x ) i = n ! δ n,k for all n, k ≥
0. The se-quence s n ( x ) is called the Sheffer sequence for ( g ( t ) , f ( t )), which is denoted by s n ( x ) ∼ ( g ( t ) , f ( t )). If s n ( x ) ∼ ( g ( t ) , t ), then s n ( x ) is called the Appell sequencefor g ( t ) (see [7,14]).Let p ( x ) ∈ P . Then we note that h f ( t ) | xp ( x ) i = h ∂ t f ( t ) | p ( x ) i = h f ′ ( t ) | p ( x ) i , (15)and h e yt − | p ( x ) i = p ( y ) − p (0) , (see [7,14]) . (16)Let us assume that s n ( x ) ∼ ( g ( t ) , f ( t )). Then we have h ( t ) = ∞ X k =0 h h ( t ) | s k ( x ) i k ! g ( t ) f ( t ) k , h ( t ) ∈ F , (17) p ( x ) = ∞ X k =0 h g ( t ) f ( t ) k | p ( x ) i k ! s k ( x ) , p ( x ) ∈ P , (18) 1 g ( ¯ f ( t )) e y ¯ f ( t ) = ∞ X k =0 s k ( y ) k ! t k , for all y ∈ C p , (19)where ¯ f ( t ) is the compositional inverse of f ( t ), and f ( t ) s n ( x ) = ns n − ( x ) , (see [7,14]) . (20)As is well known, the Euler polynomials are defined by the generating function tobe 2 e t + 1 e xt = e E ( x ) t = ∞ X n =0 E n ( x ) t n n ! , (see [1-19]) , (21)with the usual convention about replacing E n ( x ) by E n ( x ). In the special case, x = 0, E n (0) = E n are called the n -th Euler numbersLet s n ( x ) ∼ ( g ( t ) , t ). Then Appell identity is known to be s n ( x + y ) = n X k =0 (cid:18) nk (cid:19) s n − k ( x ) y k = n X k =0 (cid:18) nk (cid:19) s k ( x ) y n − k . (22)From (21), we note that the recurrence relation of the Euler numbers is given by E = 1 , ( E + 1) n + E n = E n (1) + E n = 2 δ ,n . (23)By (1) and (21), we get Z Z p ( x + y ) n dµ − ( y ) = E n ( x ) , Z Z p x n dµ − ( y ) = E n , (24)where n ≥ p -adic integrals on Z p arising from the umbral calculus. TAEKYUN KIM, DAE SAN KIM, SANGTAE JEONG AND SEOG-HOON RIM Umbral calculus and fermionic p -adic integrals on Z p Let s n ( x ) ∼ ( g ( t ) , t ). Then, by (19), we get1 g ( t ) x n = s n ( x ) if and only if x n = g ( t ) s n ( x ) . (25)Let us assume that g ( t ) = e t +12 . Then we note that g ( t ) is an invertible functional.By (21), we get 1 g ( t ) e xt = ∞ X k =0 E k ( x ) t k k ! . (26)Thus, from (26), we have1 g ( t ) x n = E n ( x ) , tE n ( x ) = ng ( t ) x n − = nE n − ( x ) . (27)By (19), (20) and (27), we see that E n ( x ) is an Appell sequence for g ( t ) = e t +12 .It is easy to show that E n +1 ( x ) = (cid:16) x − g ′ ( t ) g ( t ) (cid:17) E n ( x ) , ( n ≥ . (28)From (2), (21) and (24), we note that Z Z p e ( x + y +1) t dµ − ( y ) + Z Z p e ( x + y ) t dµ − ( y ) = 2 e xt . (29)Thus, by (29), we get Z Z p ( x + y + 1) n dµ − ( y ) + Z Z p ( x + y ) n dµ − ( y ) = 2 x n . (30)From (24) and (30), we have E n ( x + 1) + E n ( x ) = 2 x n , ( n ≥ . (31)By (28), we see that g ( t ) E n +1 ( x ) = g ( t ) xE n ( x ) − g ′ ( t ) E n ( x ) , ( n ≥ . (32)Thus, we have ( e t + 1) E n +1 ( x ) = ( e t + 1) xE n ( x ) − e t E n ( x ) . (33)By (33), we get E n +1 ( x + 1) + E n +1 ( x ) = ( x + 1) E n ( x + 1) + xE n ( x ) − E n ( x + 1) . (34)Thus, from (34) and (31), we have E n +1 ( x + 1) + E n +1 ( x ) = x ( E n ( x + 1) + E n ( x )) . (35)By (35), we get E n ( x + 1) + E n ( x ) = x ( E n − ( x + 1) + E n − ( x )) = x ( E n − ( x + 1) + E n − ( x ))= · · · = x n ( E ( x + 1) + E ( x )) = 2 x n . Let us consider the functional f ( t ) such that h f ( t ) | p ( x ) i = Z Z p p ( u ) dµ − ( u ) , (36) NOTE ON FERMIONIC p -ADIC INTEGRALS ON Z p AND UMBRAL CALCULUS 5 for all polynomials p ( x ). It can be determined from (8) to be f ( t ) = ∞ X k =0 h f ( t ) | x k i k ! t k = ∞ X k =0 Z Z p u k dµ − ( u ) t k k ! = Z Z p e ut dµ − ( u ) . (37)By (29) and (37), we get f ( t ) = Z Z p e ut dµ − ( u ) = 2 e t + 1 . (38)Therefore, by (38), we obtain the following theorem. Theorem 2.1.
For p ( x ) ∈ P , we have h Z Z p e yt dµ − ( y ) | p ( x ) i = Z Z p p ( u ) dµ − ( u ) . That is, h e t + 1 | p ( x ) i = Z Z p p ( u ) dµ − ( u ) . Also, the n -th Euler number is given by E n = h Z Z p e yt dµ − ( y ) | x n i . By (3) and (30), we get ∞ X n =0 Z Z p ( x + y ) n dµ − ( y ) t n n ! = Z Z p e ( x + y ) t dµ − ( y ) = ∞ X n =0 Z Z p e yt dµ − ( y ) x n t n n ! . (39)From (24) and (39), we have E n ( x ) = Z Z p e yt dµ − ( y ) x n = 2 e t + 1 x n , (40)where n ≥ Theorem 2.2.
For p ( x ) ∈ P , we have Z Z p p ( x + y ) dµ − ( y ) = Z Z p e yt dµ − ( y ) p ( x ) = 2 e t + 1 p ( x ) . From (22), we note that E n ( x + y ) = n X k =0 (cid:18) nk (cid:19) E k ( x ) y n − k . The Euler polynomials of order r are defined by the generating function to be (cid:16) e t + 1 (cid:17) × (cid:16) e t + 1 (cid:17) × · · · × (cid:16) e t + 1 (cid:17)| {z } r times e xt = (cid:16) e t + 1 (cid:17) r e xt = ∞ X n =0 E ( r ) n ( x ) t n n ! . (41) TAEKYUN KIM, DAE SAN KIM, SANGTAE JEONG AND SEOG-HOON RIM
In the special case, x = 0, E ( r ) n (0) = E ( r ) n are called the n -th Euler numbers oforder r ( r ≥ g r ( t ) = (cid:16) e t +12 (cid:17) r . Then we see that g r ( t )is an invertible functional in F . By (41), we get1 g r ( t ) e xt = ∞ X n =0 E ( r ) n ( x ) t n n ! . (42)Thus, we have 1 g r ( t ) x n = E ( r ) n ( x ) , tE ( r ) n ( x ) = ng r ( t ) x n − = nE ( r ) n − ( x ) . (43)So, by (42), we see that E ( r ) n ( x ) is the Appell sequence for (cid:16) e t +12 (cid:17) r . From (22), wehave E ( r ) n ( x + y ) = n X k =0 (cid:18) nk (cid:19) E ( r ) n − k ( x ) y k . (44)It is easy to show that Z Z p · · · Z Z p e ( x + ··· x r + x ) t dµ − ( x ) · · · dµ − ( x r ) = (cid:16) e t + 1 (cid:17) r e xt = ∞ X n =0 E ( r ) n ( x ) t n n ! . (45)By (6) and (45), we get E ( r ) n = h Z Z p · · · Z Z p | {z } r − times e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) | x n i , ( n ≥ , (46)and, by (10), h Z Z p · · · Z Z p e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) | x n i = X n = i + ··· + i r (cid:18) ni , · · · , i r (cid:19) h Z Z p e x t dµ − ( x ) | x i i · · · h Z Z p e x r t dµ − ( x r ) | x i r i = X n = i + ··· + i r (cid:18) ni , · · · , i r (cid:19) E i E i · · · E i r . (47)From (46) and (47), we have E ( r ) n = X n = i + ··· + i r (cid:18) ni , · · · , i r (cid:19) E i · · · E i r . (48)By (44) and (48), we see that E ( r ) n ( x ) is a monic polynomial of degree n withcoefficients in Q . Let r ∈ N . Then we note that g r ( t ) = 1 Z Z p · · · Z Z p | {z } r − times e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) = (cid:16) e t + 12 (cid:17) r . (49) NOTE ON FERMIONIC p -ADIC INTEGRALS ON Z p AND UMBRAL CALCULUS 7
By (49), we get1 g r ( t ) e xt = Z Z p · · · Z Z p e ( x + ··· x r + x ) t dµ − ( x ) · · · dµ − ( x r ) = ∞ X n =0 E ( r ) n ( x ) t n n ! . (50)From (50), we have E ( r ) n ( x ) = Z Z p · · · Z Z p ( x + · · · x r + x ) n dµ − ( x ) · · · dµ − ( x r )= 1 Z Z p · · · Z Z p | {z } r − times e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) x n = 1 g r ( t ) x n . (51)Therefore, by (51), we obtain the following theorem. Theorem 2.3.
For p ( x ) ∈ P and r ∈ N . Then we have Z Z p · · · Z Z p | {z } r − times p ( x + · · · x r + x ) dµ − ( x ) · · · dµ − ( x r ) = (cid:16) e t + 1 (cid:17) r p ( x ) . In particular, E ( r ) n ( x ) = (cid:16) e t + 1 (cid:17) r x n = Z Z p · · · Z Z p e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) x n . That is, E ( r ) n ( x ) ∼ (cid:16) R Z p · · · R Z p e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) , t (cid:17) . Let us take the functional f r ( t ) such that h f r ( t ) | p ( x ) i = Z Z p · · · Z Z p | {z } r − times p ( x + · · · x r ) dµ − ( x ) · · · dµ − ( x r ) , (52)for all polynomials p ( x ). It can be determined from (8) to be f r ( t ) = ∞ X k =0 h f r ( t ) | x k i k ! t k = ∞ X k =0 Z Z p · · · Z Z p | {z } r − times ( x + · · · x r ) k dµ − ( x ) · · · dµ − ( x r ) t k k != Z Z p · · · Z Z p | {z } r − times e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r )(53)Therefore, by (52) and (53) , we obtain the following theorem. TAEKYUN KIM, DAE SAN KIM, SANGTAE JEONG AND SEOG-HOON RIM
Theorem 2.4.
For p ( x ) ∈ P , we have h Z Z p · · · Z Z p | {z } r − times e ( x + ··· x r ) t dµ − ( x ) · · · dµ − ( x r ) | p ( x ) i = Z Z p · · · Z Z p | {z } r − times p ( x + · · · x r ) dµ − ( x ) · · · dµ − ( x r ) . Moveover, h (cid:16) e t + 1 (cid:17) r | p ( x ) i = Z Z p · · · Z Z p | {z } r − times p ( x + · · · x r ) dµ − ( x ) · · · dµ − ( x r ) . ACKNOWLEDGEMENTS. This research was supported by Basic Science Re-search Program through the National Research Foundation of Korea(NRF) fundedby the Ministry of Education, Science and Technology 2012R1A1A2003786.
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Kim
Department of Mathematics,Kwangwoon University, Seoul 139-701, Republic of Korea
E-mail: [email protected]
Dae San
Kim
Department of Mathematics,Sogang University, Seoul 121-742, Republic of Korea
E-mail: [email protected]
Sangtae
Jeong
Department of Mathematics,Inha University, Incheon 860-7114, Republic of Korea
E-mail: [email protected]
Seog-Hoon
Rim
Department of Mathematics Education,Kyungpook National University, Taegu 702-70, Republic of Korea