aa r X i v : . [ m a t h . N T ] D ec BARNES TYPE MULTIPLE q -ZETAFUNCTIONS AND q -EULER POLYNOMIALS Taekyun Kim
Abstract.
The purpose of this paper is to present a systemic study of some familiesof multiple q -Euler numbers and polynomials and we construct multiple q -zeta func-tion which interpolates multiple q -Euler numbers at negative integer. This is a partialanswer of the open question in a previous publication( see J. Physics A: Math. Gen.34(2001)7633-7638). §
1. Introduction/ Preliminaries
Let p be a fixed odd prime number. Throughout this paper Z p , Q p , C and C p will,respectively, denote the ring of p -adic rational integers, the field of p -adic rationalnumbers, the complex number field and the completion of algebraic closure of Q p .Let v p be the normalized exponential valuation of C p with | p | p = p − v p ( p ) = p . Whenone talks of q -extension, q is variously considered as an indeterminate, a complexnumber q ∈ C or p -adic number q ∈ C p . If q ∈ C , one normally assumes | q | <
1. If q ∈ C p , one normally assumes | − q | p < . We use the notation[ x ] q = 1 − q x − q , and [ x ] − q = 1 − ( − q ) x q , (see [4, 5, 6, 7]) . The q -factorial is defined as [ n ] q ! = [ n ] q [ n − q · · · [2] q [1] q . For a fixed d ∈ N with( p, d ) = 1, d ≡ mod
2. Barnes type multiple q -Euler numbers and polynomials Let x, w , w , · · · , w r be complex numbers with positive real parts. In C , the Barnestype multiple Euler numbers and polynomials are defined by(6) 2 r Q rj =1 ( e w j t + 1) e xt = ∞ X n =0 E ( r ) n ( x | w , · · · , w r ) t n n ! , for | t | < max { π | w i | | i = 1 , · · · , r } , and E ( r ) n ( w , · · · , w r ) = E ( r ) n (0 | w , · · · , w r )(see [11, 12, 14]). In this section, we as-sume that q ∈ C p with | − q | p <
1. We first consider the q -extension of Eulerpolynomials as follows:(7) ∞ X n =0 E n,q ( x ) t n n ! = Z Z p e [ x + y ] q t dµ ( y ) = 2 ∞ X m =0 ( − m e [ m + x ] q t , (see [7, 8, 17]) . Thus, we have E n,q ( x ) = − q ) n P nl =0 ( nl ) ( − l q lx q l ( see [7]). In the special case x = 0, E n,q = E n,q (0) are called the q -Euler numbers. The q -Euler polynomials of order3 ∈ N are also defined by(8) ∞ X n =0 E ( r ) n,q ( x ) t n n ! = Z Z p · · · Z Z p e [ x + x + ··· + x r ] q t dµ ( x ) · · · dµ ( x r )= 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) ( − m e [ m + x ] q t , (see [7, 8]) . In the special case x = 0, the sequence E ( r ) n,q (0) = E ( r ) n,q are refereed as the q -extensionof the Euler numbers of order r . Let f ∈ N with f ≡ mod E ( r ) n,q ( x ) = Z Z p · · · Z Z p [ x + x + · · · + x r ] nq dµ ( x ) · · · dµ ( x r )= 2 r (1 − q ) n n X l =0 (cid:18) nl (cid:19) ( − l q lx f − X a , ··· ,a r =0 ∞ X m , ··· ,m r =0 q l ( P ri =1 ( a i + fm i ) ( − P ri =1 ( a i + fm i ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r [ m + · · · + m r + x ] nq . By (8) and (9), we obtain the following theorem.
Theorem 1.
For n ∈ Z + , we have E ( r ) n,q ( x ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r [ m + · · · + m r + x ] nq = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) ( − m [ m + x ] nq . Let F ( r ) q ( t, x ) = P ∞ n =0 E ( r ) n,q ( x ) t n n ! . Then we have(10) F ( r ) q ( t, x ) = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) ( − m e [ m + x ] q t = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r e [ m + ··· + m r + x ] q t . Let χ be the Dirichlet’s character with conductor f ∈ N with f ≡ mod q -Euler polynomials attached to χ are defined by(11) ∞ X n =0 E n,χ,q ( x ) t n n ! = 2 ∞ X m =0 ( − m χ ( m ) e [ m + x ] q t . E n,χ,q ( x ) = f − X a =0 χ ( a )( − a Z Z p [ x + a + f y ] nq dµ ( y ) = [ f ] nq f − X a =0 χ ( a )( − a E n,q f ( x + af ) . In the special case x = 0, the sequence E n,χ,q (0) = E n,χ,q are called the n -th gen-eralized q -Euler numbers attached to χ . From (2) and (3), we can easily derive thefollowing equation. E m,χ,q ( nf ) − ( − n E m,χ,q = 2 nf − X l =0 ( − n − − l χ ( l )[ l ] mq . Let us consider higher-order generalized q -Euler polynomials attached to χ as follows:(13) Z X · · · Z X r Y i =1 χ ( x i ) ! e [ x + ··· + x r + x ] q t dµ ( x ) · · · dµ ( x r ) = ∞ X n =0 E ( r ) n,χ,q ( x ) t n n ! , where E ( r ) n,χ,q ( x ) are called the n -th generalized q -Euler polynomials of order r attachedto χ . By (13), we see that(14) E ( r ) n,χ,q ( x ) = 2 r (1 − q ) n n X l =0 (cid:18) nl (cid:19) q lx ( − l f − X a , ··· ,a r =0 r Y j =1 χ ( a j ) ( − q l ) P ri =1 a i (1 + q lf ) r = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) ( − m f − X a , ··· ,a r =0 r Y j =1 χ ( a j ) ( − P ri =1 a i [ r X j =1 a j + x + mf ] nq , and(15) ∞ X n =0 E ( r ) n,χ,q ( x ) t n n ! = 2 r ∞ X m , ··· ,m r =0 ( − P rj =1 m j r Y i =1 χ ( m i ) ! e [ x + P rj =1 m j ] q t . In the special case x = 0, the sequence E ( r ) n,χ,q (0) = E ( r ) n,χ,q are called the n -th gener-alized q -Euler numbers of order r attached to χ .By (14) and (15), we obtain the following theorem. Theorem 2.
Let χ be the Dirichlet’s character with conductor f ∈ N with f ≡ mod . For n ∈ Z + , r ∈ N , we have E ( r ) n,χ,q ( x ) = 2 r (1 − q ) n n X l =0 (cid:18) nl (cid:19) q lx ( − l f − X a , ··· ,a r =0 r Y j =1 χ ( a j ) ( − q l ) P ri =1 a i (1 + q lf ) r = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) ( − m f − X a , ··· ,a r =0 r Y j =1 χ ( a j ) ( − P ri =1 a i [ r X j =1 a j + x + mf ] nq = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r r Y i =1 χ ( m i ) ! [ x + m + · · · + m r ] nq . For h ∈ Z and r ∈ N , we introduce the extended higher-order q -Euler polynomialsas follows:(16) E ( h,r ) n,q ( x ) = Z Z p · · · Z Z p q P rj =1 ( h − j ) x j [ x + x + · · · + x r ] nq dµ ( x ) · · · dµ ( x r ) , (see [8]) . From (16), we note that(17) E ( h,r ) n,q ( x ) = 2 r ∞ X m , ··· ,m r =0 q ( h − m + ··· +( h − r ) m r ( − m + ··· + m r [ x + m + · · · + m r ] nq . It is known in [8] that(18) E ( h,r ) n,q ( x ) = 2 r (1 − q ) n n X l =0 (cid:0) nl (cid:1) ( − q x ) l ( − q h − r + l ; q ) r = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) q ( − q h − r ) m [ x + m ] nq . Let F ( h,r ) q ( t, x ) = P ∞ n =0 E ( h,r ) n,q ( x ) t n n ! . Then we have(19) F ( h,r ) q ( t, x ) = 2 r ∞ X m =0 (cid:18) m + r − m (cid:19) q q ( h − r ) m ( − m e [ m + x ] q t = 2 r ∞ X m , ··· ,m r =0 q P rj =1 ( h − j ) m j ( − P rj =1 m j e [ x + m + ··· + m r ] q t . Therefore, we obtain the following theorem.
Theorem 3.
For h, ∈ Z , r ∈ N , and x ∈ Q + , we have E ( h,r ) n,q ( x ) = 2 r ∞ X m , ··· ,m r =0 q ( h − m + ··· +( h − r ) m r ( − m + ··· + m r [ m + · · · + m r + x ] nq . f ∈ N with f ≡ mod E ( h,r ) n,q ( x ). E ( h,r ) n,q ( x ) = [ f ] nq f − X a , ··· ,a r =0 ( − a + ··· + a r q P rj =1 ( h − j ) a j E n,q f ( x + a + · · · + a r f ) . Let us consider Barnes’ type multiple q -Euler polynomials. For w , · · · , w r ∈ Z p , wedefine the Barnes’ type q -multiple Euler polynomials as follow:(20) E ( r ) n,q ( x | w , · · · , w r ) = Z Z p · · · Z Z p [ r X j =1 w j x j + x ] nq dµ ( x ) · · · dµ ( x r ) . From (20), we can easily derive the following equation.(21) E ( r ) n,q ( x | w , · · · , w r ) = 2 r (1 − q ) n n X l =0 (cid:0) nl (cid:1) ( − q x ) l (1 + q lw ) · · · (1 + q lw r ) , (see [8] ) . Thus, we have(22) E ( r ) n,q ( x | w , · · · , w r ) = 2 r (1 − q ) n n X l =0 (cid:18) nl (cid:19) ( − q x ) l f − X a , ··· ,a r =0 ( − P ri =1 a i q l P rj =1 w j a j (1 + q lfw ) · · · (1 + q lfw r ) , where f ∈ N with f ≡ mod E ( r ) n,q ( x | w , · · · , w r ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r [ x + w m + · · · + w r m r ] nq . In the special case x = 0, the sequence E ( r ) n,q ( w , · · · , w r ) = E ( r ) n,q (0 | w , · · · , w r ) arecalled the n -th Barnes’ type multiple q -Euler numbers. Let F ( r ) q ( t, x | w , · · · , w r ) = P ∞ n =0 E ( r ) n,q ( x | w , · · · , w r ) t n n ! . Then we have(24) F ( r ) q ( t, x | w , · · · , w r ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r e [ x + w m + ··· + w r m r ] q t . Therefore we obtain the following theorem.
Theorem 4.
For w , · · · , w r ∈ Z p , r ∈ N , and x ∈ Q + , we have E ( r ) n,q ( x | w , · · · , w r ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r [ x + m w + · · · + m r w r ] nq = 2 r (1 − q ) n n X l =0 (cid:0) nl (cid:1) ( − q x ) l (1 + q lw ) · · · (1 + q lw r ) . w , · · · , w r ∈ Z p , a , · · · , a r ∈ Z , we consider another q -extension of Barnes’type multiple q -Euler polynomials as follows:(25) E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) = Z Z p · · · Z Z p [ x + r X j =1 w j x j ] nq q P ri =1 a i x i r Y i =1 dµ ( x i ) ! . Thus, we have(26) E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) = 2 r (1 − q ) n n X l =0 (cid:0) nl (cid:1) ( − l q lx (1 + q lw + a ) · · · (1 + q lw r + a r ) . From (25) and (26), we can derive the following equation.(27) E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) = 2 r X m , ··· ,m r =0 ( − P rj =1 m j q P ri =1 a i m i [ x + r X j =1 w j x j ] nq . Let F ( r ) q ( t, x | w , · · · , w r ; a , · · · , a r ) = P ∞ n =0 E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) t n n ! . Then,we have(28) F ( r ) q ( t, x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r q a m + ··· + a r m r e [ x + w m + ··· + w r m r ] q t . Theorem 5.
For r ∈ N , w , · · · , w r ∈ Z p , and a , · · · , a r ∈ Z , we have E ( r ) n,q ( x | w , · · · , w r ; a , · · · a r ) = 2 r ∞ X m , ··· ,m r =0 ( − P rj =1 m j q P ri =1 a i m i [ x + r X j =1 w j m j ] nq . Let χ be a Dirichlet’s character with conductor f ∈ N with f ≡ mod q -multiple Euler polynomials attached to χ as follows: E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r )= Z X · · · Z X [ x + w x + · · · + w r x r ] nq r Y j =1 χ ( x j ) q a x + ··· + a r x r dµ ( x ) · · · dµ ( x r ) . Thus, we have(29) E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r )= 2 r (1 − q ) n f − X b , ··· ,b r =0 r Y i =1 χ ( b i ) ! ( − P rj =1 b j q P ri =1 ( lw i + a i ) b i P nl =0 (cid:0) nl (cid:1) ( − l q lx Q rj =1 (1 + q ( lw j + a j ) f ) . E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 r Y j =1 χ ( m i ) ( − m + ··· + m r q a m + ··· + a r m r [ x + r X j =1 w j m j ] nq . Therefore we obtain the following theorem.
Theorem 6.
For r ∈ N , w , · · · , w r ∈ Z p , and a , · · · , a r ∈ Z , we have E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 r Y j =1 χ ( m i ) ( − m + ··· + m r q a m + ··· + a r m r [ x + r X j =1 w j m j ] nq . Let F ( r ) q,χ ( t, x | w , · · · , w r ; a , · · · , a r ) = P ∞ n =0 E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r ) t n n ! . By Theorem 6, we see that(30) F ( r ) q,χ ( t, x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 r Y j =1 χ ( m i ) ( − m + ··· + m r q a m + ··· + a r m r e [ x + P rj =1 w j m j ] q t . §
3. Barnes type multiple q -zeta functions In this section, we assume that q ∈ C with | q | < w , · · · , w r are positive. From (28), we consider the Barnes’ type multiple q -Euler polynomialsin C as follows:(31) F ( r ) q ( t, x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r q a m + ··· + a r m r e [ x + w m + ··· + w r m r ] q t = ∞ X n =0 E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) t n n ! , for | t | < max ≤ i ≤ r { π | w i | } . For s, x ∈ C with ℜ ( x ) > , a , · · · , a r ∈ C , we can derive the following Eq.(32) fromthe Mellin transformation of F ( r ) q ( t, x | w , · · · , w r ; a , · · · , a r ).(32) 1Γ( s ) Z ∞ t s − F ( r ) q ( − t, x | w , · · · , w r ; a , · · · , a r ) dt = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r q m a + ··· + m r a r [ x + w m + · · · + w r m r ] sq . s, x ∈ C with ℜ ( x ) > a , · · · , a r ∈ C , we define Barnes’ type multiple q -zetafunction as follows:(33) ζ q,r ( s, x | w , · · · , w r ; a , · · · , a r ) = 2 r ∞ X m , ··· ,m r =0 ( − m + ··· + m r q m a + ··· + m r a r [ x + w m + · · · + w r m r ] sq . Note that ζ q,r ( s, x | w , · · · , w r ) is meromorphic function in whole complex s -plane.By using the Mellin transformation and the Cauchy residue theorem, we obtain thefollowing theorem which is a part of answer of open question in [6, p.7637 ] . Theorem 7.
For x ∈ C with ℜ ( x ) > , n ∈ Z + , we have ζ q,r ( − n, x | w , · · · , w r ; a , · · · , a r ) = E ( r ) n,q ( x | w , · · · , w r ; a , · · · , a r ) . Let χ be a Dirichlet’s character with conductor f ∈ N with f ≡ mod q -Euler polynomials attachedto χ in C as follows:(34) F ( r ) q,χ ( t, x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 r Y j =1 χ ( m i ) ( − m + ··· + m r q a m + ··· + a r m r e [ x + P rj =1 w j m j ] q t = ∞ X n =0 E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r ) t n n ! . From (34) and Mellin transformation of F ( r ) q,χ ( t, x | w , · · · , w r ; a , · · · , a r ), we caneasily derive the following equation (35) .(35) 1Γ( s ) Z ∞ t s − F ( r ) q,χ ( − t, x | w , · · · , w r ; a , · · · , a r ) dt = 2 r ∞ X m , ··· ,m r =0 (cid:16)Q rj =1 χ ( m i ) (cid:17) ( − m + ··· + m r q m a + ··· + m r a r [ x + w m + · · · + w r m r ] sq . For s, x ∈ C with ℜ ( x ) >
0, we also define Barnes’ type multiple q - l -function asfollows:(36) l ( r ) q,χ ( s, x | w , · · · , w r ; a , · · · , a r )= 2 r ∞ X m , ··· ,m r =0 (cid:16)Q rj =1 χ ( m j ) (cid:17) ( − m + ··· + m r q m a + ··· + m r a r [ x + w m + · · · + w r m r ] sq . Note that l ( r ) q,χ ( s, x | w , · · · , w r ) is meromorphic function in whole complex s -plane.By using (34), (35), (36), and the Cauchy residue theorem, we obtain the followingtheorem. 10 heorem 8. For x, s ∈ C with ℜ ( x ) > , n ∈ Z + , we have l ( r ) q,χ ( − n, x | w , · · · , w r ; a , · · · , a r ) = E ( r ) n,χ,q ( x | w , · · · , w r ; a , · · · , a r ) . We note that Theorem 8 is r -multiplication of Dirichlet’s type q - l -series. Theorem8 seems to be interesting and worthwhile for doing study in the area of multiple p -adic l -function or mathematical physics related to Knot theory and ζ -function (see [4-20]). References [1] I. N. Cangul,V. Kurt, H. Ozden, Y. Simsek,
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