Some identities of symmetry for q-Euler polynomials under the symmetric group of degeree n arising from fermionic p-adic q-integrals on Zp
aa r X i v : . [ m a t h . N T ] A p r SOME IDENTITIES OF SYMMETRY FOR q -EULERPOLYNOMIALS UNDER THE SYMMETRIC GROUP OFDEGREE n ARISING FROM FERMIONIC p -ADIC q -INTEGRALSON Z p DMITRY V. DOLGY, DAE SAN KIM, AND TAEKYUN KIM
Abstract.
In this paper, we investigate some new symmetric identities forthe q -Euler polynomials under the symmetric group of degree n which arederived from fermionic p -adic q -integrals on Z p . Introduction
Let p be a fixed prime number such that p ≡ Z p , Q p and C p will denote the ring of p -adic integers, the field of p -adicrational numbers and the completion of the algebraic closure of Q p . Let q be anindeterminate in C p such that | − q | p < p − p − . The p -adic norm is normalized as | p | p = p and the q -analogue of the number x is defined as [ x ] q = − q x − q . Note thatlim q → [ x ] q = x .As is well known, the Euler numbers are defined by E = 1 , ( E + 1) n + E n = 2 δ ,n , ( n ∈ N ∪ { } ) , with the usual convention about replacing E n by E n (see [1–14]).The Euler polynomials are given by E n ( x ) = n X l =0 (cid:18) nl (cid:19) x n − l E l = ( E + x ) n , ( n ≥ , (see [1, 2]) . In [8], Kim introduced Carlitz-type q -Euler numbers as follows:(1.1) E ,q = 1 , q ( q E q + 1) n + E n,q = [2] q δ ,n , ( n ≥ , (see [8]) , with the usual convention about replacing E nq by E n,q .The Carlitz-type q -Euler polynomials are also defined as(1.2) E n,q ( x ) = (cid:16) q x E q + [ x ] q (cid:17) n = n X l =0 (cid:18) nl (cid:19) q lx E l,q [ x ] n − lq , (see [3, 8]) . Let C ( Z p ) be the space of all C p -valued continuous functions on Z p . Then, for f ∈ C ( Z p ), the fermionic p -adic q -integral on Z p is defined by Kim as I − q ( f ) = Z Z p f ( x ) dµ − q ( x )(1.3) Mathematics Subject Classification.
Key words and phrases.
Identities of symmetry, Carlitz-type q -Euler polynomial, Symmetricgroup of degree n , Fermionic p -adic q -integral. = lim N →∞ p N ] − q p N − X x =0 f ( x ) ( − q ) x = lim N →∞ q q p N p N − X x =0 f ( x ) ( − q ) x , (see [5–11]) . From (1.3), we note that(1.4) q n I − q ( f n ) + ( − n − I − q ( f ) = [2] q n − X l =0 ( − n − − l f ( l ) , ( n ∈ N ) , (see [8]) . The Carlitz-type q -Euler polynomials can be represented by the fermionic p -adic q -integral on Z p as follows:(1.5) E n,q ( x ) = Z Z p [ x + y ] nq dµ − q ( y ) , ( n ≥ , (see [8]) . Thus, by (1.5), we get E n,q ( x )(1.6) = n X l =0 (cid:18) nl (cid:19) q lx Z Z p [ y ] lq dµ q ( y ) [ x ] n − lq = n X l =0 (cid:18) nl (cid:19) q lx E l,q [ x ] n − lq , (see [8]) . From (1.4), we can easily derive(1.7) q Z Z p [ x + 1] nq dµ − q ( x ) + Z Z p [ x ] nq dµ − q ( x ) = [2] q δ ,n , ( n ∈ N ∪ { } ) . The equation (1.7) is equivalent to(1.8) q E n.q (1) + E n,q = [2] q δ ,n , ( n ≥ . The purpose of this paper is to give some new symmetric identities for theCarlitz-type q -Euler polynomials under the symmetric group of degree n which arederived from fermionic p -adic q -integrals on Z p .2. Symmetric identities for E n,q ( x ) under S n Let w , w , . . . , w n ∈ N such that w ≡ w ≡ w ≡ · · · ≡ w n ≡ Z Z p e ( Q n − j =1 w j ) y + ( Q nj =1 w j ) x + w n P n − j =1 Q n − i =1 i = j w i k j q t dµ − q w ··· wn − ( y )(2.1) = lim N →∞ p N ] q w ··· wn − × p N − X y =0 e ( Q n − j =1 w j ) y + ( Q nj =1 w j ) x + w n P n − j =1 Q n − i =1 i = j w i k j q t ( − q w ··· w n − ) y YMMETRY FOR q -EULER POLYNOMIALS 3 = 12 lim N →∞ [2] q w ··· wn − × w n − X m =0 p N − X y =0 e ( Q n − j =1 w j ) ( m + w n y )+ ( Q nj =1 w j ) x + w n P nj =1 Q n i =1 i = j w i k j q t × ( − m + y q w ··· w n − ( m + w n y ) . Thus, by (2.1), we get1[2] q w ··· wn − n − Y l =1 w l − X k l =0 ( − P n − i =1 k i q w n P n − j =1 Q n − i =1 i = j w i k j (2.2) × Z Z p e ( Q n − j =1 w j ) y + ( Q nj =1 w j ) x + w n P n − j =1 Q n − i =1 i = j w i k j q t dµ − q w ··· wn − ( y )= 12 lim N →∞ n − Y l =1 w l − X k l =0 w n − X m =0 p N − X y =0 ( − P n − i =1 k i + m + y × q w n P n − j =1 Q n − i =1 i = j w i k j + ( Q n − j =1 w j ) m + ( Q nj =1 w j ) y × e ( Q n − j =1 w j ) ( m + w n y )+ ( Q nj =1 w j ) x + w n P n − j =1 Q n − i =1 i = j w i k j q t . As this expression is invariant under any permutation σ ∈ S n , we have thefollowing theorem. Theorem 2.1.
Let w , w , . . . , w n ∈ N such that w ≡ w ≡ · · · ≡ w n ≡ . Then, the following expressions q wσ (1) ··· wσ ( n − n − Y l =1 w σ ( l ) − X k l =0 ( − P n − i =1 k i q w σ ( n ) P n − j =1 Q n − i =1 i = j w σ ( i ) k j × Z Z p e ( Q n − j =1 w σ ( j ) ) y + ( Q nj =1 w j ) x + w σ ( n ) P n − j =1 Q n − i =1 i = j w σ ( i ) k j q t dµ q wσ (1) ··· wσ ( n − ( y ) are the same for any σ ∈ S n , ( n ≥ . Now, we observe that n − Y j =1 w j y + n Y j =1 w j x + w n n − X j =1 n − Y i =1 i = j w j k j q t (2.3) = n − Y j =1 w j q y + w n x + w n n − X j =1 k j w j q w ··· wn − . DMITRY V. DOLGY, DAE SAN KIM, AND TAEKYUN KIM
By (2.3), we get Z Z p e ( Q n − j =1 w j ) y + ( Q nj =1 w j ) x + w n P n − j =1 Q n − i =1 i = j w i k j q t dµ − q w ··· wn − ( y )(2.4)= ∞ X m =0 n − Y j =1 w j mq Z Z p y + w n x + w n n − X j =1 k j w j mq w ··· wn − dµ − q w ··· wn − ( y ) t m m != ∞ X m =0 n − Y j =1 w j mq E m,q w ··· wn − w n x + w n n − X j =1 k j w j t m m ! . For m ≥
0, from (2.4), we have Z Z p n − Y j =1 w j y + n Y j =1 w j x + w n n − X j =1 n − Y i =1 i = j w i k j mq dµ − q w ··· wn − ( y )(2.5)= n − Y j =1 w j mq E m,q w ··· wn − w n x + w n n − X j =1 k j w j , ( n ∈ N ) . Therefore, by Theorem 2.1 and (2.5), we obtain the following theorem.
Theorem 2.2.
Let w , . . . w n ∈ N be such that w ≡ w ≡ · · · ≡ w n ≡ .For m ≥ , the following expressions hQ n − j =1 w σ ( j ) i mq [2] q wσ (1) ··· wσ ( n − n − Y l =1 w σ ( l ) − X k l =0 ( − P n − i =1 k i q w σ ( n ) P n − j =1 Q n − i =1 i = j w σ ( i ) k j × E m,q wσ (1) ··· wσ ( n − w σ ( n ) x + w σ ( n ) m − X j =1 k j w σ ( j ) are the same for any σ ∈ S n . It is not difficult to show that y + w n x + w n n − X j =0 k j w j q w ··· wn − (2.6)= [ w n ] q hQ n − j =1 w j i q n − X j =1 n − Y i =1 i = j w i k j q wn + q w n P n − j =1 Q n − i =1 i = j w i k j [ y + w n x ] q w ··· wn − . YMMETRY FOR q -EULER POLYNOMIALS 5 Thus, by (2.6), we get Z Z p y + w n x + w n n − X j =0 k j w j mq w ··· wn − dµ q − w ··· wn − ( y )(2.7)= m X l =0 (cid:18) ml (cid:19) [ w n ] q hQ n − j =1 w j i q m − l n − X j =1 n − Y i =1 i = j w i k j m − lq wn q lw n P n − j =1 Q n − i =1 i = j w i k j × Z Z p [ y + w n x ] lq w ··· wn − dµ − q w ··· wn − ( y )= m X l =0 (cid:18) ml (cid:19) [ w n ] q hQ n − j =1 w j i q m − l n − X j =1 n − Y i =1 i = j w i k j m − lq wn × q lw n P n − j =1 Q n − i =1 i = j w i k j E l,q w ··· wn − ( w n x ) . From (2.7), we have hQ n − j =1 w j i mq [2] q w ··· wn − n − Y l =1 w l − X k l =0 ( − P n − l =1 k l q w n P n − j =1 Q n − i =1 i = j w i k j × Z Z p y + w n x + w n n − X j =1 k j w j nq w ··· wn − dµ − q w ··· wn − ( y )(2.8)= m X l =0 (cid:18) ml (cid:19) hQ n − j =1 w j i lq [2] q w ··· wn − [ w n ] m − lq E l,q w ··· wn − ( w n x ) × n − Y s =1 w s − X k s =0 ( − P n − j =1 k j q ( l +1) w n P n − j =1 Q n − i =1 i = j w i k j n − X j =1 n − Y i =1 i = j w i k j m − lq wn = 1[2] q w w ··· wn − m X l =0 (cid:18) ml (cid:19) n − Y j =1 w j lq [ w n ] m − lq E l,q w ··· wn − ( w n x ) × ˆ T m,q wn ( w , w , . . . , w n − | l ) , where ˆ T m,q ( w , . . . , w n − | l )(2.9) DMITRY V. DOLGY, DAE SAN KIM, AND TAEKYUN KIM = n − Y s =1 w s − X k s =0 q ( l +1) P n − j =1 Q n − i =1 i = j w i k j n − X j =1 n − Y i =1 i = j w i k j m − lq ( − P n − j =1 k j . As this expression is invariant under any permutation in S n , we have the follow-ing theorem. Theorem 2.3.
Let w , w , . . . , w n ∈ N be such that w ≡ w ≡ · · · ≡ w n ≡ . For m ≥ , the following expressions q wσ (1) wσ (2) ··· wσ ( n − m X l =0 (cid:18) ml (cid:19) n − Y j =1 w σ ( j ) lq (cid:2) w σ ( n ) (cid:3) m − lq × E l,q wσ (1) ··· wσ ( n − (cid:0) w σ ( n ) x (cid:1) ˆ T m,q wσ ( n ) (cid:0) w σ (1) , w σ (2) , . . . , w σ ( n − | l (cid:1) are the same for any σ ∈ S n . Acknowledgements. This paper is supported by grant NO 14-11-00022 of RussianScientific Fund.
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