Some Weighted Sum Formulas for Multiple Zeta, Hurwitz Zeta, and Alternating Multiple Zeta Values

We perform a further investigation for the multiple zeta values and their variations and generalizations in this paper. By making use of the method of the generating functions and some connections between the higher-order trigonometric functions and the Lerch zeta function, we explicitly evaluate some weighted sums of the multiple zeta, Hurwitz zeta, and alternating multiple zeta values in terms of the Bernoulli and Euler polynomials and numbers. It turns out that various known results are deduced as special cases.

It is well known that one of the central problems on the multiple zeta values is to determine all the possible Q-linear relations among them. Goncharov's ( [5], Conjecture 4.2) conjecture implies that it suffices to study the relations among the multiple zeta values of the same weight |a| � α 1 + α 2 + · · · + α k . Perhaps the earliest result in this direction is Euler's [6] sum formula, namely, α 1 +α 2 �n ζ α 1 , α 2 + 1 � ζ(n + 1), n ≥ 2, where ζ(·) is the Riemann zeta function. In fact, there exists a general form of (2), as follows: |a|�n ζ α 1 , α 2 , . . . , α k + 1 � ζ(n + 1), n ≥ k, which is referred to as the "sum conjecture" in [7], and was proved by Granville [8] and Zagier [9] independently around the year 1995. Some new relations for the multiple zeta values and their different variations and generalizations have been found in recent years. For example, Ohno and Zudilin [10] in 2008 proved a weighted form of Euler's sum formula: α 1 +α 2 �n 2 α 2 +1 ζ α 1 , α 2 + 1 � (n + 2)ζ(n + 1), n ≥ 2. (4) It becomes obvious from (2) that Ohno and Zudilin's weighed sum formula can be rewritten as Based on the equivalence of (4) and (5), Guo and Xie [11] in 2009 extended the weighted sum formula of Ohno and Zudilin to arbitrary depth and discovered that for positive integers n, k with n ≥ k, where S j � α k− j+1 + · · · + α k + 1 for j � 1, . . . , k − 1. For another weighted sum formulas of the multiple zeta values, one is referred to [12], where the weight coefficients are given by (symmetric) polynomials of the arguments. On the contrary, let m, n, k be positive integers with m ≥ 2 and n ≥ k, and let E(mn, k) be the sums of all multiple zeta values of the depth k and the weight mn given by E(mn, k) � |a|�n ζ mα 1 , . . . , mα k .
Gangl et al. [13] in 2004 proved that Shen and Cai [14] in 2012 obtained the formulas After that, Hoffman [15] used the theory of symmetric functions to establish the general sum formula where B n is the n-th Bernoulli number. Furthermore, Zhao [16] used the ideas developed in [15] to evaluate the sums of all multiple Hurwitz zeta values of the depth k and the weight 2n in terms of the Euler numbers. Moreover, Zhao [17] used the theory of symmetric functions to consider the more complicated alternating multiple zeta values and depicted that the sums of all alternating multiple zeta values of the depth k and the weight 2n can be evaluated in terms of the Riemann zeta function and the Euler numbers. More recently, Chen et al. [18] used the method of the generating functions to express E(mn, k) by constructing a combinatorial identity of products of the multiple zeta values and the so-called multiple zeta-star values at the repetitions of m, and then used Muneta's [19] and Nakamura's [20] results to reobtain Hoffman's sum formula (10) and confirm Genčev's ( [21], Conjecture 4.1) conjecture on the evaluation of E(4n, k).
Subsequently, Shen and Jia [22] extended the sums of the multiple Hurwitz zeta values previously considered in [23] and showed that the sums of all multiple Hurwitz zeta values of the depth k and the weight mn can be expressed by a combinatorial identity of products of the multiple Hurwitz zeta values and the so-called multiple Hurwitz zeta-star values at the repetitions of m. In particular, Shen and Jia [22] obtained Zhao's [16] sum formula with a slight different notation and evaluated the sums of all multiple Hurwitz zeta values of the depth k and the weight 4n in terms of the Euler numbers.
Motivated and inspired by the work of the above authors, we explicitly evaluate some weighted sums of the multiple zeta, Hurwitz zeta, and alternating multiple zeta values in terms of the Bernoulli and Euler polynomials and numbers by using the method of the generating functions and some connections between the higher-order trigonometric functions and the Lerch zeta function established by the first author [24]. e results presented here are the corresponding extensions of various known sum formulas. is paper is organized as follows: In Section 2, we give several weighted sum formulas for the multiple zeta values, some of which generalize the sum formulas (10) and (11), and improve Eie and Ong's [25] weighted sum formulas. In Section 3, we present some similar weighted sum formulas for the multiple Hurwitz zeta values and deduce Shen and Jia's [22] sum formulas as special cases. Section 4 concentrates on the features that have contributed to the weighted sum formulas for the alternating multiple zeta values, and it then turns out that Zhao's [17] sum formula is obtained in a rather simple way.

Sum Formulas for Multiple Zeta Values
For convenience, in the following, we always denote by i the square root of − 1 such that i 2 � − 1, s(n, k) the Stirling numbers of the first kind, B n (x) the Bernoulli polynomials, and E n (x) the Euler polynomials. It is clear that taking x � 0 and x � 1/2 in the Bernoulli and Euler polynomials gives the Bernoulli numbers B n � B n (0) and the Euler numbers E n � 2 n E n (1/2), respectively. We refer the reader to two standard books [26,27] on basic properties for these special 2 Journal of Mathematics sequences and polynomials. We also write [t n ]f(t) as the coefficients of t n in f(t) for nonnegative integer n. We now state our first result as follows.
and U(m, n, k) is the linear combination of the Bernoulli polynomials given for positive integer m and nonnegative integers n, k by Proof. Recall that Euler's infinite product formula of the sine function is which holds true for arbitrary complex number x (see [26], p. 75 or [28], pp. [12][13][14][15][16][17][18]. e binomial series asserts that for complex number α (see [27], p. 37), where α n are the binomial coefficients given for non- (17) So from (15) and (16), we discover that for complex number x with 0 < |x| < 1, Comparing the coefficients of x 2n on both sides of (18), it then follows that for complex number x with 0 < |x| < 1, (19) We now evaluate the right-hand side of (19) from another view. Let q, n be positive integers, and let θ r be a real function defined on positive integer r. If θ r ≠ 0, ± q, ± 2q, . . ., then (see [24], eorem 3.2) where ϕ(a, x, s) is the Lerch zeta function given for real number a, x ≠ negative integer or zero, and complex number s by Note that the series is an entire function of s when a is not an integer. Obviously, replacing n by m and θ r /q by x in (20) gives that for real number x ≠ 0, ± 1, ± 2, . . ., (22) It follows from (19) and (22) (23) It is easily seen from the Taylor series expansion for the complex exponential function and the familiar binomial theorem that Since for nonnegative integer n (see [29], eorem 12.13), where ζ(s, x) is the Hurwitz zeta function given for real number x > 0 and complex number s by so by applying (25) to (24), we arrive at Now (12) follows from (23) and (27). is completes the proof of eorem 1.
Corollary 1 is usually attributed to Hoffman ([15], Corollary 2) and was previously obtained by Aoki,Kombu,and Ohno ([30], Equation (4.6)), who stated it in the language of the multiple zeta-star values. We are in a good position to use eorem 1 to yield the following result. □ Theorem 2. Let m, n, k be positive integers with n ≥ k. en where V 1 (m, n) is a rational number given for positive integer m and nonnegative integer n by Proof. Clearly, for real or complex parameter λ, Just as a polynomial function of z in the order of ascending power is divided by another polynomial function of z in the order of ascending power, we discover that Applying (33) to the right-hand side of (32), it then follows from (18)

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Noticing that from (19) and eorem 1, we have We now evaluate the coefficients of x 2n− 2l in the infinite product of the right-hand side of (34). Let μ 1 (λ), . . . , μ m (λ) be all complex numbers determined by the factorization of the polynomial function 1 − λ(1 − z) m + λ over the complex number field satisfying that e famous Vieta's theorem implies that So from (36), (37), and the remarkable formula see [31], Equation (36), or [7], Corollary 2.3, where a { } n denotes the n repetitions of a, we get that Inserting (35) and (39) into (34), it follows that us (30) follows immediately after making k-times derivative with respect to λ and then taking λ � − 1 on both sides of (40). is concludes the proof of eorem 2.
It is easy to check that taking m � 1 in eorem 2 and then applying (29) and s(1, 1) � 1 leads to Hoffman's formula (10). It is worth noticing that the formula (40) can also be regarded as an extension of eorem 1. In a similar consideration to eorem 2, we have the following result.
where V 2 (m, n) is a rational number given for positive integer m and nonnegative integer n by and B (m) n (x) are the higher-order Bernoulli polynomials defined by the generating function (see [32]):

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Proof. We know from (16) and (33) that If we replace x by ix in (15), then we have It follows from (15), (35), and (45) that On the contrary, from (36), (37), and the well known formula (see [31], Equation (37)), we obtain that 6 Journal of Mathematics Inserting (46) and (48) erefore, we get (41) and finish the proof of eorem 3 when making k-times derivative with respect to λ and then taking λ � − 1 on both sides of (49).
It is trivial to check that Genčev's conjecture (11) holds true when taking m � 1 in eorem 3. We here remark that eorems 2 and 3, as well as the generalizations of the sum formulas (10) and (11), are the improvements of the results recently obtained by Eie and Ong ( [25], eorems 2.1 and 2.2), where they expressed the left-hand side of (30) by a combinatorial identity involving the higher derivative of one function and the sums of products of m Bernoulli polynomials, and the left-hand side of (41) by a combinatorial identity involving the higher derivative of another function and the sums of products of 2m Bernoulli polynomials. □

Sum Formulas for Multiple Hurwitz Zeta Values
In this section, we shall study the multiple Hurwitz zeta values defined by the series (see [23,33]) where α 1 , α 2 , . . . , α k are all positive integers with α k ≥ 2 and present some weighted sum formulas for them. e results showed here are very analogous to the ones in Section 2.

Theorem 4. Let m, n be positive integers. en
where U(m, n, k) is the linear combination of the Euler polynomials given for positive integer m and nonnegative integers n, k by Proof. It is well known that Euler's infinite product formula of the cosine function is which holds true for arbitrary complex number x (see [26], p. 75 or [28], pp. [12][13][14][15][16][17][18]. Hence, we obtain from (16) and (53) that for complex number x with |x| < 1/2, (54) If we replace x by 1/2 − x in (22), we get that for real number x ≠ ±(1/2), ±(3/2), . . ., Journal of Mathematics which can be converted by the expression of the higherorder secant function stated in ( [24], eorem 3.2). It then follows that Observe that Since for nonnegative integer n (see [34], Corollary 3), where η(s, x) is the alternating Hurwitz zeta function (also called Hurwitz Euler-eta function) given for real number x > 0 and complex number s by so by applying (58) to (57), we find that Inserting (60) into (56), we have us, we complete the proof of eorem 4 by equating (54) and (61).
In what follows, we denote by T(mn, k) the sums of all multiple Hurwitz zeta values of the depth k and the weight mn given for positive integers m, n, k with m ≥ 2 and n ≥ k by It follows that we state the following result.
□ Corollary 2. Let n be a positive integer. en Proof. By taking m � 1 in eorem 4, in view of s(1, 1) � 1 and E n � 2 n E n (1/2) for nonnegative integer n, the desired result follows immediately. We mention that Corollary 2 was obtained by Shen and Jia ( [22], p. 265) in the language of the multiple Hurwitz zeta-star values. We now give another weighted sum formula for the multiple Hurwitz zeta values as follows.
where V 3 (m, n) is a rational number given for positive integer m and nonnegative integer n by
is completes the proof of eorem 5. □ Corollary 3. Let n, k be positive integers with n ≥ k. en Proof. Taking m � 1 in eorem 5, in light of s(1, 1) � 1 and E n � 2 n E n (1/2) for nonnegative integer n, we get the desired result. Corollary 3 is a general form of Shen and Cai's [23] results for the cases 2 ≤ k ≤ 5 in T(2n, k) and was also found by Shen where V 4 (m, n) is a rational number given for positive integer m and nonnegative integer n by and E (m) n (x) are the higher-order Euler polynomials defined by the generating function (see [35]) Proof. With the help of (16) and (33), we discover that By replacing x by ix in (53), we find that It follows from (53), (67), and (76) that 10 Journal of Mathematics If we apply (36), (37), and the formula (see [22], p. 265), to the second infinite product in the right-hand side of (75), we have Inserting (77) and (79) into (75), it then follows that Journal of Mathematics us we prove (72) immediately by making k-times derivative with respect to λ and then taking λ � − 1 on both sides of (80). is concludes the proof of eorem 6.
Proof. Since E n (x) � E (1) n (x) for nonnegative integer n, so by taking m � 1 in eorem 6, the desired result follows from s(1, 1) � 1 and E n � 2 n E n (1/2) for nonnegative integer n.
Proof. By substituting x/2 for x in (15) and xi/2 for x in (53), we find that It follows from (16) and (84) that Trivially, the left-hand side of (85) can be written as