COSINE HIGHER-ORDER EULER NUMBER CONGRUENCES AND DIRICHLET L-FUNCTION VALUES

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1 Kyushu J. Math , doi:10.06/kyushujm COSINE HIGHER-ORDER EULER NUMBER CONGRUENCES AND DIRICHLET L-FUNCTION VALUES Nianliang WANG, Hailong LI and Guodong LIU Received 0 Aril 016 and revised 9 Setember 016 Abstract. In this aer we obtain the residue modulo a rime ower of cosine higher-order Euler numbers H k n m in terms of the linear combination of the Dirichlet L-function values Ls, χ at ositive integral arguments s or of generalized Bernoulli numbers. Our results are restricted to the equal arity case; i.e. s and χ are of the same arity. In the rocess, we emloy Yamamoto s results on finite exressions in terms of Dirichlet L-function values for short interval character sums and in this sense our treatment is decisive, i.e. any ad-hoc transformation of short interval sums. The results obtained not only generalize the revious results ertaining to the congruences modulo a rime ower of the class numbers as the secial case of s = 1 in terms of Euler numbers but also closes the chater on ossible similar research. 1. Introduction Secial values of zeta- and L-functions have been a central object of research starting from Euler s solution to the Basler roblem, which refers to the evaluation of the infinite series n=1 1/n, which in turn is the value of the Riemann zeta-function ζsat s =. In analogy to the Riemann zeta-function, the Dirichlet L-function associated with a Dirichlet character χ mod q is defined in the first instance by Ls, χ = χnn s 1.1 n=1 and then continued analytically meromorhically in the case of rincial characters over the whole lane by the functional equation. As a generalization of Euler s formula for ζn, n being even ositive integers, it is known that Ln, χ = 1 n δ+/ πn τχ i δ n!q n B n, χ 1. cf. e.g. [1, 6] if n δmod, whereδ being 0 or 1 is defined as the arity χ 1 = 1 δ. Here, τχ= χae πia/q a mod q 010 Mathematics Subject Classification: Primary 11A07, 11A15, 11L0, 11B68, 11S0. Keywords: generalized Bernoulli number; cosine higher-order Euler numbers; weighted short-interval character sum; Dirichlet class number formula; Dirichlet L-function. c 017 Faculty of Mathematics, Kyushu University

2 198 N. Wang et al is the normalized Gauss sum and B n,χ = B n χ are the generalized Bernoulli numbers attached to χ which we refer to as χ-bernoulli numbers subsequently cf. [, 3]. They are most conveniently defined by q a B n,χ = q n 1 χab n, 1.3 q a=1 where B n x is the nth Bernoulli olynomial. For the sake of comleteness, we state the generating function for the χ-bernoulli olynomials B n,χ x of which χ-bernoulli numbers are the secial values at x = 0: q a=1 χateat e qt e xt = 1 n=0 B n,χ x tn n!, t < π q. Thinking of the multilicative function χ n = 1foralln 0χ 0 = 0 to be the even trivial character modulo 1, 1. amounts to the well-known Euler s formula for ζn. The value at s = 1 is meaningless in the case of the Riemann zeta-function but the value L1,χ at s = 1 can be made meaningful as a conditionally convergent series and is of utmost imortance in whole number theory as a main ingredient of the class number formula. We confine ourselves to the imaginary quadratic field Q d with discriminant d<0. Let hd denote its class number and let χ d a = a d be the corresonding Kronecker character. Then Dirichlet s class number formula reads hd = w d π L1,χ d, 1. where w is the number of roots of 1 in Q d. This is the first stage of the class number formula. Thesecond stage consists ofexressingtheinfinite series forl1,χin finite form, i.e. as a short interval character sum. We recall that there has been a lot of work done to exress short-intervalcharactersums in terms of Dirichlet L-function values. This is indeed a reverse roblem to the second stage. The roblem of exressing short-interval character sums in terms of Ln, χ has been settled by Yamamoto [] and in rincile, any short-interval character sum can be exressed in Dirichlet L-function values. In articular, for Q, 1 mod, 1. reads h = π L1,χ = B 1 χ, 1.5 where the second equality is a finite exression for the infinite series L1,χcf. [5]. Equation 1.5 gives an exression for the most imortant class number in terms of a rather simle number by way of L1,χ. Note that χ indicates the real rimitive Dirichlet character mod associated with the Gaussian field Qi and that the χ-bernoulli number associated with χ is essentially the Euler number E n = 1 n + 1 B n+1,χ, n 0; we have an exression for the class number h = 1 in terms of the first Euler number. Since there are higher-order Euler numbers E n which are introduced as a secial case of cosine Euler numbers by 1.7 below, it may be of some interest to study their numbertheoretic roerties, in articular congruences modulo a ower of a rime. In this regard,

3 Cosine higher-order Euler number congruences 199 Zhang and Xu deduced the result [6, ] on the ϕ α /th Euler number E ϕ α / with 1 mod : E ϕ α / 1/ s=1 s = i π τχ L1,χ mod α, 1.6 whichisequaltoh by the class number formula cf. [5, 7]. Since the 1/th sum can be exressed as a multile of L1,χ, 1.6 can be interreted as an exression of the Euler number in terms of the multile of L1,χ. Several authors derived similar results on the subject, e.g. cf. [3 10]. In view of the allowance summand l our results suersede most of the revious results. We introduce the cosine higher-order Euler numbers H k n m by cf. [8] n=1 H k tn n m n! = sec t cos mtk 1.7 and write H n m = H 1 n m. 1.8 It follows that 1 n H k n 0 = Ek n, Ek n being the nth higher-order Euler number, and that E n = E 0 n are ordinary Euler numbers. In this aer we are concerned with congruences for cosine Euler numbers H n m and H n m in terms of L-function values, or what amounts to the same thing, χ- Bernoulli numbers modulo a ower of a rime; i.e. we find secific linear combinations which are congruent to the cosine Euler numbers. We will do this via the weighted shortinterval character sums Sr,N k χ.13. Our Lemma 1 and Lemma are modified forms of Yamamoto s dee result and Lemma will serve for exressing short-interval sums in terms of χ-bernoulli numbers. We fix the following convention. Convention. As in [7] we assume the following for the index n: ϕ α n = β + l, β 1.9 denoting β γ = ϕ α /, 1.10 and that l = n β lies in the range 0 l< ϕα Throughout, in what follows, we let be an odd rime and let α, ν N. Since our argument starts by changing the ower of a natural number by the Legendre symbol raised to the ower α, highly non-trivial results are obtained only for γ being an odd integer. However, in view of the allowance summand l we may also obtain somewhat non-trivial results for γ being even.

4 00 N. Wang et al If γ is an odd integer l = 0, 3.1 with ν = 1 reads H n 1 n i π τχ L1,χ = 1 n h mod α, mod, which generalizes 1.6. In the case 1 mod, the Legendre symbol χ a = a is the real rimitive even Dirichlet character to the modulus associated with the real quadratic field Q and the Kronecker symbol χ a = χ χ a = χ a a is the real rimitive odd character associated with Q. On the other hand, in the case 3 mod, the Legendre symbol is the real rimitive odd character associated with the imaginary quadratic field Q and the Kronecker symbol χ aχ a = χ a a is the real rimitive even character associated with the real quadratic field Q,. Preliminaries PROPOSITION 1. cf. [8] In the notation above, and letting N 0 = N {0}, we have H k n m + 1 = In articular, Proof. Noting that ν 0,ν 1,...,ν m N 0 ν 0 +ν 1 + +ν m =k H k! 1 n+k+ν 0+ν 1 + +m+1ν m m H n m + 1 = 1 n n m + 1 = 1n ν 0!ν 1! ν m! m m jν j n..1 m 1 m j j n,. 1 j m + 1 jj n..3 sec t cosm + 1t = eim+1t + e im+1t e it + e it = = we may rewrite 1.7 as H k tn n m n! = n=1 m n=0 n=1 ν 0,ν 1,...,ν m N 0 ν 0 +ν 1 + +ν m =k m m 1 n+j m j n n!, n t n m jν j n! 1 j e im jt tn k! 1 n+k+ν 0+ν 1 + +m+1ν m ν 0!ν 1! ν m!. by multinomial exansion. Comaring the coefficients of t n /n! on both sides of., we conclude.1.

5 Cosine higher-order Euler number congruences 01 PROPOSITION. Under the convention, we have m j γ H n m m+n 1 j j l mod α,.5 H n m + 1 1n m 1 j m + 1 jj l j γ mod α..6 Proof. By the Euler Fermat theorem and the Euler criterion, for any integer a, one has a a ϕα / mod α..7 Raising.7 to the ower γ,wehave, j β = j ϕα / γ j γ mod α..8 Hence, relacing n by β + l on the right-hand side of. and then relacing the owers j β by the Legendre symbol in.8 leads to.5. Similarly, we have.6. We have As immediate consequences, for any ositive integer m,wehave where E n is the nth Euler number For a rime 1 mod, we have H n+ϕ α m + 1 H n m + 1 mod α,.9 H n+ϕ α m + 1 H n m + 1 mod α..10 H n α 1 n E n mod α,.11 H ϕ α / 0 mod α..1 We define the weighted short-interval character sum associated with χ modulo M with olynomial weight as Sr,N k χ := Sk r,0,um χ = 1 N k χaa k,.13 0 a N/r where N is a ositive multile of M and r are ositive integers such that N and r are relatively rime, and where the rime on the summation sign means that for extremal values of a, the corresonding summand is to be halved. As is well known, S1,N k χ and χ Bernoulli olynomials are related to each other by cf. [9, ]or[10, 1.] k + 1N k S1,N k N 1 χ = k + 1 χaa k = B k+1,χ N B k+1,χ..1 a=1 The following lemmas disclose a close relationshi between the weighted short-interval sums and generalized Bernoulli numbers or the linear combination of Dirichlet L-series. Lemma 1 was derived by the first author in [7], which is a slight generalization of Yamamoto s result cf. [, 5.1, 5.] for the characters rimitive modulo and q is a ower of.

6 0 N. Wang et al LEMMA 1. [7, Lemma 1.3] Let 0 <r Z,k N {0}, q = ν, and let χ be a rimitive character mod. Then Sr,q k k+1 χ = k!τχ πi a ν 1 a 1 r k a+1 k a + 1! a=1 n=1 b a ν 1 nχn n a,.15 where b a n = 1 a+1 χ 1η n η n 1 a k,.16 b k+1 n = 1 k+1 χ 11 η n + 1 η n, η = e πi/r. This is [7, Lemma 1.3] with β = 1/r. LEMMA. [1, 6]or[11,.1] Let N be a ositive multile of M, sayn = um, and let χ be a Dirichlet character modulo M.Letr be a ositive integer rime to N. We have Sr,N k 1 B k+1,χ χ = N k k χr rn k ϕr k+1 1 k + 1 ψ N N k+1 a B a,χψ,.17 k + 1 a ψ a=0 where the sum ψ is over all Dirichlet characters ψ modulo r. Proof. The roof of.17 follows immediately from [1, 6]: k + 1r k 0 a N/r χaa k = B k+1,χ r k + χr ϕr and k+1 k + 1 B k+1,χψ N = N k+1 a B a,χψ, a a=0 or directly from [11,.1]. ψ NB k+1,χψ N ψ 3. Main results In this section we resent our main results, forming the weighted short-interval character sums to a linear combination of generalized Bernoulli numbers, or a linear combination of Dirichlet L-functions, in terms of H k n m rime ower modulo congruences. For some more results on this asect, readers may refer to Carlitz [19], Yamamoto [], Szmidt et al [1], Wang et al [7], Kanemitsu et al [11]andalsoto[9 18] Congruences related to H n m THEOREM 1. Assume the convention. Then, if γ, is odd and l is even, we have, for 1 mod, l/ 1 β+l/ H β+l ν 1 j j+1 lν jν 1 l!τχ π j+1 Lj + 1,χ χ mod α, l j! 3.1

7 and, for 3 mod and l is odd, 1 β+l/ H β+l ν l+1/ Cosine higher-order Euler number congruences 03 1 j 1 i j+1 lν j 1ν 1 l!τχ π j Lj, χ χ mod α. 3. l j + 1! If γ is even, we have, for 1 mod, H β+l ν β+l/ l+1 1 l + 1 and, for 3 mod, { l ν + 3 B l+1 l l ν B l+1 } + l 1 l+1 1B l+1 H β+l ν β+l/ l+1 1 l + 1 ν + 1 B l+1 B l+1 ν { l ν + 3 B l+1 l l ν B l+1 } + 1 ν l 1 l+1 1B l+1 mod α, 3.3 ν + 1 B l+1 B l+1 ν mod α. 3. Proof. The roof is in the sirit of that of [7, Theorem.1]. In the case where γ is odd, we use j in lace of j γ in.5. We deduce from Proosition that H β+l ν l+1 1 ν 1/+β+l/ [ ν 1/] l+1 j l j ν 1/ j l j mod α. 3.5 For 1 mod and l even, we have 1 ν 1/ = 1, whence 3.5 reads H β+l ν l+1 1 β+l/ lν l+1 S, l ν χ χ S, l ν χ mod α. 3.6 By Lemma 1 we have S, l ν χ l/ 1 j l!τχ χ = l j+1 π j ν 1 j 1 l j + 1! j j Lj, χ and l/ 1 j l!τχ + π j+1 l j ν 1 j l j! Lj + 1,χ χ, 3.7 S, l ν χ l/ 1 j l!τχ = π j l ν 1 j 1 1 χ l j + 1! j Lj, χ. 3.8 Substituting 3.7 and 3.8 in 3.6, we conclude 3.1.

8 0 N. Wang et al For 3 mod and l odd we have 1 ν 1/ = 1 ν, whence 3.5 reads H β+l ν l+1 1 β+l/+ν lν l+1 S, l ν χ χ S, l ν χ mod α. 3.9 By Lemma 1 again we have l+1/ S, l ν χ 1 j+ν 1 il!τχ = π j l j+1 ν 1j 1 l j + 1! Lj, χ χ l 1/ + 1 j l!τχ l j+1 ν 1 j π j+1 il j! χ j 1 j Lj + 1,χ 3.10 and l 1/ S, l ν χ 1 j l!τχ = l ν 1 j π j+1 1 χ il j! j Lj + 1,χ Substituting 3.10 and 3.11 in 3.9, we conclude 3.. In the case where γ is even, we have the rincile Dirichlet character [1] j in lace of j γ in.5, whence l being even, we have H β+l ν 1 ν 1/+β+l/ which by arity argument reduces to H β+l ν 1 ν 1/+β+l/ [ { ν 1/] l+ [ ν 1/] ν 1/ j l [1] j l+1 1 j j l [1] j mod α, The sum of the tye N j l [1] j can be ut in closed form by N } j l [1] j mod α. j l [1] j = 1 [ ] N B l+1 N + 1 l B l l 1B l l + 1 Noting the dulication formula x x + 1 B l+1 x = l B l+1 + l B l+1, 3.13 we may ut N =[ ν 1/] or ν 1/ andx = ν + 1/ or ν 1 + 1/, as the case may be, and distinguishing the cases 1or3mod, we obtain 3.3 or 3., comleting the roof. Remark 1. We remark that Theorem 1 can be treated as a slight generalization of [7, Theorem.1]; i.e., when ν = α Theorem 1 above reads [7, Theorem.1]. As an immediate corollary, we have the following.

9 Cosine higher-order Euler number congruences 05 COROLLARY 1. Let ν be a ositive integer. If n/ϕ α / is an odd integer, then we have H n ν 1 n i π τχ χ L1,χ χ = 1 n h mod α 3.1 for 1 mod.if n/ϕ α / is an even integer, then we have H n ν mod α, , 1 mod H n ν 0, 3 mod, ν even mod α n, 3 mod, ν odd. By Lemma, Theorem 1 may be also exressed as follows. THEOREM. Assume the convention and further assume that γ is odd. Then we have l/ H β+l ν 1 β+l/ l νl jb j+1,χ j j + 1 mod α, 3.17 for 1 mod, and l+1/ H β+l ν 1 β+l/ l νl j+1 B j,χ j 1 j mod α, 3.18 for 3 mod. 3.. Congruences related to H n m THEOREM 3. Assume the convention. If γ is odd, we have H β+l H β+l ν 1 β+l/ l+ ν + 1 l/ 1 j+1 l!j 1 νl+ j+j 1 τχ π j l j +! 1 χ Lj, χ + 1 β/ l+3 l+1 l + 1!τχ j π l+ 1 χ Ll +,χ mod α, 3.19 l+ l/ 1 β+l/ l!τχ νl 1 j j+1 π j+1 l + 1 j! ν 1j j ν + 1Lj + 1,χ + 1β/ l+1 τχ l + 1! π l+ l+ χ Ll +,χ mod α 3.0

10 06 N. Wang et al for 1 mod and l even, and H β+l H β+l ν 1 β+l/ l+3 l!τχ νl+1 ν + 1 l 1/ 1 j+1 j π j+1 il j + 1! ν 1j 1 χ j+1 Lj + 1,χ + 1β+1/ l+3 l + 1!τχ l+1 π l+ i 1 χ l+ Ll +,χ mod α, 3.1 l+1/ 1 β+l/ l!τχ νl for 3 mod lodd and ν even. 1 j i j π j l j +! ν 1j 1 j ν 1Lj, χ + 1β+1/ l + 1!τχ l+1 π l+ i l+ χ Ll +,χ mod α 3. If γ is even, we have H β+l ν 1 β+l/ l+1 ν l ν + 1 ν B l+1 B l+1 l + 1 l l ν ν 1 B l+1 B l+1 + l+1 1 l 1B l+1 for 1 mod and H β+l β+l/ l+1 1 l + l+1 l+1 B l+ ν l+1 ν + 1 B l+ ν B l+ ν 1 B l+ + l+ 1 l+1 1B l+ mod α 3.3 ν β+l/ l ν + 1 l ν + 3 ν + 1 B l+1 B l+1 l / l l ν ν B l+1 B l+1 + l+1 1 l 1B l+1 β+l/ l+1 1 l+1 ν + 3 ν + 1 B l+ B l+ l +

11 Cosine higher-order Euler number congruences / l+1 l+1 ν ν B l+ B l+ + l+ 1 l+1 1B l+ mod α 3. is valid for 1 mod, ν being ositive integers, or for 3 mod,andν even. Proof. The roof goes along similar lines to that of Theorem 1, and we state only the case of γ being odd. Corresonding to 3.5, we have H ν 1 j ν 1 j β+l ν 1 β+l/ ν 1 j j l 1 j j l+1 mod α 3.5 or after some transformations, H β+l ν 1 β+l/ l+1 νl+1 S, l ν χ l+1 νl+1 S1, l ν χ l+1 νl+1 S, l+1 ν χ + l+1 νl+1 S l+1 1, ν χ mod α. 3.6 For 1 mod, l being even, by Lemma 1, we have l/ 1 j+1 l + 1!τχ 1, ν χ = π j j ν 1 j 1 l j +! Lj, χ, 3.7 S l+1 S l+1 l/ 1 j l + 1!τχ, ν χ = π j l+ ν 1 j 1 1 χ l j +! j Lj, χ + 1l+/ l + 1!τχ π l+ l ν 1 l+1 1 χ l+ Ll +,χ, 3.8 S1, l ν χ l/ 1 j+1 l!τχ = j 1 π j l j + 1! ν 1j 1 Lj, χ. 3.9 Substituting 3.7, 3.8 and 3.9 in to 3.6 we have For 3 mod, l being odd, by Lemma 1 again, we have l 1/ S l+1 1 j+1 l + 1!τχ 1, ν χ = π j+1 j i ν 1 j l j + 1! Lj + 1,χ, 3.30 l 1/ S l+1, ν χ = 1 j l + 1!τχ l+1 ν 1 j π j+1 1 χ il j + 1! j Lj + 1,χ + 1l+1/ l + 1!τχ l+1 ν 1 l+1 π l+ i 1 χ l+ Ll +,χ, 3.31 l 1/ S1, l ν χ 1 j+1 l!τχ = j π j+1 il j! ν 1jLj + 1,χ. 3.3 Substituting 3.30, 3.31 and 3.3 in to 3.6 we have 3.1.

12 08 N. Wang et al Similarly, for 1 mod, or 3 mod, ν even, we have ν 1 mod, and therefore ν + 1/ being odd, we have ν β+l/ ν + 1 νl l+1 S, l ν χ l χs, l ν χ H β+l l+1 νl+1 S l+1, ν χ + l+1 νl+1 χs l+1, ν χ mod α, 3.33 by distinguishing χ = χ or χ. From Lemma 1, l/ 1 j l + 1!τχ χ, ν χ = l j+ π j ν 1 j 1 l j +! j j Lj, χ S l+1 l/ 1 j l + 1!τχ + π j+1 l j+1 ν 1 j l j + 1! Lj + 1,χ + 1l+/ l + 1!τχ l+ π l+ ν 1 l+1 + χ l+ l+ Ll +,χ, 3.3 l+1/ S l+1 1 j 1 il + 1!τχ, ν χ = π j l j+3 ν 1 j 1 l j +! Lj, χ l 1/ + 1 j l + 1!τχ l j+3 π j+1 ν 1 j il j + 1! Lj + 1,χ χ j 1 j + 1l+1/ l + 1!τχ π l+ i ν 1 l+1 + χ l+1 1 l+1 Ll +,χ Substituting 3.8, 3.9, 3.3 and 3.7 or 3.31, 3.3, 3.35 and 3.10 in to 3.33 we have 3.19 or 3.. Acknowledgements. The authors would like to exress their hearty thanks to Professor Shigeru Kanemitsu for roviding them with the direction of this research and for enlightening discussions. The first author would like to thank the anonymous referee for his/her valuable suggestions and corrections. The work was Suorted by the Natural Science Basic Research Project of Shaanxi Province of China Program No. 016JM103 and by the Shangluo Science Research lan Program No. SK and by the Science research roject of Shaanxi Provincial Deartment of Education Program No. 16JM165 and 16JK138. REFERENCES [1] S. Kanemitsu, J. Urbanowicz and N. L. Wang. On some new congruences for generalized Bernoulli numbers. Acta Arith , 7 58, doi:10.06/aa [] K. Chen. Sums of the roducts of generalized Bernoulli olynomials. Pacific J. Math ,

13 Cosine higher-order Euler number congruences 09 [3] T. Kim, K. Hwang and Y. Kim. Symmetry roerties of higher-order Bernoulli olynomials. Adv. Difference Equ , Article ID , doi: /009/ [] Y. Yamamoto. Dirichlet series with eriodic coefficients. Proc. Intern. Symos. Algebraic Number Theory Kyoto, JSPS, Tokyo, [5] N. L. Wang and T. Arai. Class number formula for certain imaginary quadratic fields. Pure Al. Math. J , 1 6, doi: /j.amj.s [6] W.-P. Zhang and Z.-F. Xu. On a conjecture of the Euler numbers. J. Number Theory , [7] N.-L. Wang, J.-Z. Li and D.-S. Liu. Euler number congruences and Dirichlet L-functions. J. Number Theory , [8] G.-D. Liu. Generating functions and generalized Euler numbers. Doctoral Thesis, Kinki University, 006, [9] T. Kim. Symmetry roerties of the generalized higher-order Euler olynomials. Proc. Jangjeon Math. Soc , [10] N. Wang. C. Li and H. Li. Some identities on the generalized higher-order Euler and Bernoulli numbers. ARS Combinatoria, , [11] S. Kanemitsu, H. L. Li and N. L. Wang. Weighted short-interval character sums. Proc. Amer. Math. Soc , [1] J. Szmidt, J. Urbanowicz and D. Zagier. Congruences among generalized Bernoulli numbers. Acta Arith , [13] G.-D. Liu and W.-P. Zhang. Alications of an exlicit formula for the generalized Euler numbers. Acta Math. Sin. Engl. Ser. 008, [1] G.-D. Liu. Generating functions and generalized Euler numbers. Proc. Jaan Acad. Ser. A, Math. Sci , 9 3. [15] H. Joris. On the evaluation of Gaussian sums for non-rimitive Dirichlet characters. Enseign. Math , [16] T. M. Aostol. Introduction to Analytic Number Theory. Sringer, Berlin, [17] B. C. Berndt. Classical theorems on quadratic residues. Enseign. Math. 1976, [18] A. Schinzel, J. Urbanowicz and P. van Wamelen. Class numbers and short sums of Kronecker symbols. J. Number Theory , 6 8. [19] L. Carlitz. Arithmetic roerties of generalized Bernoulli numbers. J. Reine Angew. Math , Nianliang Wang Institute of Alied Mathematics Shangluo University Shangluo, Shaanxi P.R. China wangnianliangshangluo@aliyun.com Hailong Li Deartment of Mathematics Weinan Normal University Weinan P.R. China lihailong@wnu.edu.cn Guodong Liu Deartment of Mathematics Huizhou University Huizhou Guandong P.R. China gdliu@ub.huizhou.gd.cn