IDENTITIES BY GENERALIZED L−SUMMING METHOD

M. HASSANI AND Z. JAFARI

Abstract. In this paper, we introduce 3-dimensional L−summing method, which is a rearrangement of the summation ∑ A_{abc} with 1 ≤ a, b, c ≤ n. Applying this method on some special arrays, we obtain some identities on the Riemann zeta function and digamma function. Also, we give a Maple program for this method to obtain identities with input various arrays and out put identities concerning some elementary functions and hypergeometric functions. Finally, we introduce a further generalization of L−summing method in higher dimension spaces.

1. Introduction and Motivation

Consider the following n × n multiplication table

Figure 1. Multiplication table and L−summing element, L_{k}

If we set Σ(n) for the sum of all numbers in it, then by summing line by line we have Σ(n) = (\frac{n(n+1)}{2})^{2}. On the other hand, we can find Σ(n) by using another method; letting L_{k} be the sum of numbers in the rotated L in above table (right part of Figure 1), we have

L_{k} = k + 2k + ⋯ + k^{2} + ⋯ + 2k + k = 2k(1 + 2 + ⋯ + k) − k^{2} = k^{3}.

We call L_{k}, L−summing element. Thus we get Σ(n) = ∑_{k=1}^{n} L_{k} = ∑_{k=1}^{n} k^{3}, and therefore ∑_{k=1}^{n} k^{3} =

(\frac{n(n+1)}{2})^{2}. This is 2-dimensional L-summing method (applied on the array A_{ab} = ab), which briefly is

(1.1)  ∑(L − Summing Elements) = Σ.

2000 Mathematics Subject Classification. 65B10. Key words and phrases. L−Summing Method.

More precisely, the L−summing method of elements of n × n array A_{ab} with 1 ≤ a, b ≤ n, is the following rearrangement

∑_{k=1}^{n} {∑_{a=1}^{k} A_{ak} + ∑_{b=1}^{k} A_{kb} − A_{kk}} = ∑_{1≤a,b≤n} A_{ab}.

This method allows to obtain easily some classical algebraic identities and also, with help of MAPLE, some new compact formulas for sums related with the Riemann zeta function, the gamma function and the digamma function [2, 3]. In this paper we introduce a 3-dimensional version of L−summing method for n × n × n arrays and applying it on some special arrays we obtain some identities concerning the Riemann zeta function and digamma function. Then, we give a Maple program for this method and using it we generate and then proof some new identities, concerning some elementary functions and hypergeometric functions. Finally, we introduce a further generalization of L−summing method in higher dimension spaces and for latices related by a manifold.

2. Formulation of the L−summing method in ℝ^{3}

Consider a three dimensional array A_{abc} with 1 ≤ a, b, c ≤ n and n is a positive integer. We should prepare an explicit version of the general formulation (1.1) for this array. The summation of all entries is Σ(n) = ∑_{1≤a,b,c≤n} A_{abc}. The L−summing elements in this array have the form pictured bellow

Figure 2. L−summing elements in ℝ^{3}

So, we have L_{k} = Σ_{2} − Σ_{1} + Σ_{0}, with

Σ_{2} = ∑_{b,c=1}^{k} A_{kbc} + ∑_{a,c=1}^{k} A_{akc} + ∑_{a,b=1}^{k} A_{abk},  Σ_{1} = ∑_{a=1}^{k} A_{akk} + ∑_{b=1}^{k} A_{kbk} + ∑_{c=1}^{k} A_{kkc},  Σ_{0} = A_{kkk}.

Therefore, L−summing method in ℝ^{3} take the following formulation

(2.1)  ∑_{k=1}^{n}{Σ_{2} − Σ_{1} + Σ_{0}} = Σ(n).

Note that Σ_{2} is the sum of entries in three faces, Σ_{1} is the sum of entries in three intersected edges and Σ_{0} is the end point of all faces and edges.

If the array A_{abc} is symmetric, that is for each permutation σ ∈ S_{3} it satisfies A_{abc} = A_{σ(a)σ(b)σ(c)}, then L−summing elements in ℝ^{3} takes the following easier form

(2.2)  L_{k} = 3 ∑_{1≤b,c≤k} A_{kbc} − 3 ∑_{a=1}^{k} A_{akk} + A_{kkk}.

In the next section we will apply 3-dimensional L−summing method on two special symmetric arrays, related by the Riemann zeta function and digamma function.

3. Arrays related by the Riemann zeta function and digamma function

3.1. The Riemann zeta function. Suppose s ∈ ℂ and let A_{abc} = (abc)^{−s}. It is clear that

Σ(n) = ∑_{1≤a,b,c≤n}(abc)^{−s} = (∑_{k=1}^{n} \frac{1}{k^{s}} )^{3} = ζ_{n}^{3}(s),

where ζ_{n}(s) = ∑_{k=1}^{n} k^{−s}. Since this array is symmetric, considering (2.2), we have

L_{k} = 3\frac{ζ_{k}^{2}(s)}{k^{s}} − 3\frac{ζ_{k}(s)}{k^{2s}} + \frac{1}{k^{3s}} ,

and an easy simplifying, we can reform ∑ L_{k} = Σ(n) as follows

(3.1)  ∑_{k=1}^{n} \frac{ζ_{k}^{2}(s)}{k^{s}} − \frac{ζ_{k}(s)}{k^{2s}} = \frac{ζ_{n}^{3}(s) − ζ_{n}(3s)}{3}.

Note that if ℜ(s) > 1, then lim_{n→∞} ζ_{n}(s) = ζ(s), where ζ(s) = ∑_{k=1}^{∞} n^{−s} is the well-known Riemann zeta function defined for complex values of s with ℜ(s) > 1 and admits a meromorphic continuation to whole complex plan [5]. So, for ℜ(s) > 1 we have

∑_{k=1}^{∞} \frac{ζ_{k}^{2}(s)}{k^{s}} − \frac{ζ_{k}(s)}{k^{2s}} = \frac{ζ^{3}(s) − ζ(3s)}{3},

which also is true for other values of s by meromorphic continuation, except s = 1 and s = \frac{1}{3}.

3.2. Digamma function. Setting s = 1 in (3.1) (or equivalently taking A_{abc} = \frac{1}{abc}) and considering ζ_{n}(1) = H_{n} = ∑_{k=1}^{n} \frac{1}{k}, we obtain

∑_{k=1}^{n} \frac{H_{k}^{2}}{k} − \frac{H_{k}}{k^{2}} = \frac{H_{n}^{3} − ζ_{n}(3)}{3}.

One can state this identity in sense of digamma function Ψ(x) = \frac{d}{dx} ln Γ(x), with Γ(x) = ∫_{0}^{∞} e^{−t}t^{x−1}dt is the well-known gamma function. Considering logarithmic derivative of the formula Γ(n + 1) = nΓ(n), we obtain

(3.2)  Ψ(n + 1) = \frac{1}{n} + Ψ(n),

and applying this relation, we yield that Ψ(n + 1) − Ψ(1) = H_{n}. Thus, we have

(3.3)  Ψ(n + 1) + γ = H_{n},

in which γ = 0.57721 ⋯ is the Euler constant [1]. Therefore, we obtain

(3.4)  ∑_{k=1}^{n} \{\frac{(Ψ(k + 1) + γ)^{2}}{k} − \frac{Ψ(k + 1) + γ}{k^{2}} \} = \frac{(Ψ(n + 1) + γ)^{3} − ζ_{n}(3)}{3}.

Letting

𝔖(m, n) = ∑_{k=1}^{n} \frac{Ψ(k)^{m}}{k} ,

the following identity [3] is a result of 2-dimensional L−summing method

(3.5)  𝔖(1, n) = \frac{(Ψ(n + 1) + γ)^{2} + Ψ(1, n + 1)}{2} − \frac{π^{2}}{12} − Ψ(n + 1)γ − γ^{2},

where Ψ(m, x) = \frac{d^{m}}{dx^{m}} Ψ(x) is called m^{th} polygamma function [1] and we have ∑_{k=1}^{n} \frac{1}{k^{2}} = −Ψ(1, n + 1) + \frac{π^{2}}{6} , which is special case of the following identity

(3.6)  ζ_{n}(s) = \frac{(−1)^{s−1}}{(s − 1)!} Ψ(s − 1, n + 1) + ζ(s)  (s ∈ ℤ, s ≥ 2),

and using it in (3.1) one can get a generalization of (3.4), however this relation itself is the key of getting an analogue of (3.5) in ℝ^{3}, stated bellow.

Theorem 3.1. For every integer n ≥ 1, we have

∑_{k=1}^{n} \frac{Ψ(k)^{2}}{k} + \frac{Ψ(k)}{k^{2}} = \frac{(Ψ(n + 1) + γ)^{3}}{3} − \frac{ζ_{n}(3)}{3} + (γ − 2)\frac{π^{2}}{6} − (γ − 2)Ψ(1, n + 1) − γ^{2}Ψ(n + 1) − γ^{3} − 2𝔖(1, n).

Proof. We begin from the left hand side of the identity (3.4); simplifying it by the relations (3.2), (3.3) and the relation (3.6) with s = 2, gives the result. □

Corollary 3.2. For every integer n ≥ 1, we have

𝔖(2, n) = \frac{(Ψ(n + 1) + γ)^{3}}{3} − \frac{ζ_{n}(3)}{3} + (γ − 2)\frac{π^{2}}{6} − (γ − 2)Ψ(1, n + 1) − γ^{2}Ψ(n + 1) − γ^{3} − 2𝔖(1, n) − ∑_{k=1}^{n} \frac{Ψ(k)}{k^{2}}.

In above corollary, the main term in the right hand side is \frac{Ψ(n+1)^{3}}{3} . Also, we note that the summation ∑_{k=1}^{n} \frac{Ψ(k)}{k^{2}} is converges. Thus, we can write the following asymptotic relation

𝔖(2, n) = \frac{Ψ(n + 1)^{3}}{3} + O(ln^{2} n) (n → ∞).

Similarly, considering (3.5) we have

𝔖(1, n) = \frac{Ψ(n + 1)^{2}}{2} + O(ln n) (n → ∞).

Note and Problem. It is interesting to find an explicit (probably recurrence) relation for the function 𝔖(m, n). Considering two above asymptotic relations, we guess that

𝔖(m, n) = \frac{Ψ(n + 1)^{m+1}}{m + 1} + O(ln^{m} n) (n → ∞).

One can attack to this problem considering generalization of L−summing method in higher dimension spaces, pointed in the last section of this paper.

4. Generating some new identities by Maple and 3-dimension L−summing method

Appendix of this paper includes Maple program of 3-dimension L−summing method. By LSMI < A_{abc} >, we call the identity outputted by L−Summing method’s Maple program with input A_{abc}. The algorithm of this program is result of the formulation of 3-dimension L−summing method in above sections. In this program we input a 3-dimensional array A_{abc}, then out put is an identity generated by Maple. In this section we will state some of these identities, with handling a detailed proof.

4.1. Some elementary functions.

Proposition 4.1. We have

LSMI < ln(a) >: ∑_{k=1}^{n} {k^{2} ln k + 2 k ln Γ(k + 1) − 2 k ln k − ln Γ(k + 1) + ln k} = n^{2} ln Γ(n + 1).

Proof. Considering the relations (2.1) and Γ(n + 1) = n!, we have Σ(n) = n^{2} ∑_{a=1}^{n} ln a = n^{2} ln Γ(n + 1). Also, Σ_{2} = k^{2} ln k + 2k ln Γ(k + 1), Σ_{1} = ln Γ(k + 1) + 2k ln k and Σ_{0} = ln k. Putting these relations in (2.1) yields LSMI < ln(a) > as desired. □

Corollary 4.2. We have

∑_{k=1}^{n} {(k^{2} − k) ln k + 2k ln Γ(k + 1)} = (n^{2} + n) ln Γ(n + 1).

Proof. Breaking up the statement under the summation obtained from LSMI < ln(a) > in Proposition 4.1, into the sum of (k^{2} − k) ln k + 2k ln Γ(k + 1) and ln Γ (k + 1) + k ln k − ln k, and considering the Proposition 6 of [3], which states

∑_{k=1}^{n} {ln Γ (k + 1) + k ln k − ln k} = n ln Γ(n + 1),

completes the proof. □

Remark 4.3. Examining Maple code of expressed summation on above corollary, one can see that Maple has no comment on the computing this summation; however, it is obtained by Maple itself and L−summing method. This example shows that program-writers of Maple can add L−summing method in the summation package of this software, in order to making it able to compute some summations which already couldn’t compute them.

Proposition 4.4. A little simplifying LSMI < tan(a) >, we have

∑_{k=1}^{n} {(k − 1)^{2} tan k + (2k − 1)𝔗(k)} = n^{2}𝔗(n),

where 𝔗(n) = ∑_{k=1}^{n} tan k.

Proof. Considering the relation (2.1), we have Σ(n) = n^{2} ∑_{a=1}^{n} tan a = n^{2}𝔗(n). Also, Σ_{2} = k^{2} tan k+2k𝔗(k), Σ_{1} = 𝔗(k) + 2k tan k and Σ_{0} = tan k, and consequently L_{k} = (k − 1)^{2} tan k + (2k − 1)𝔗(k). This completes the proof. □

4.2. Hypergeometric functions. In the next proposition, we introduce an identity concerning hypergeometric functions, denoted in Maple by hypergeom([a_{1} a_{2} ⋯ a_{p}],[b_{1} b_{2} ⋯ b_{q}],x). Standard notation and definition [6] is as follows

pF_{q} [ \begin{matrix}a_{1} & a_{2} & ⋯ & a_{p} \\ b_{1} & b_{2} & ⋯ & b_{q} \end{matrix} ; x] = ∑_{k≥0} t_{k}x^{k},

where

\frac{t_{k+1}}{t_{k}} = \frac{(k + a_{1})(k + a_{2}) ⋯ (k + a_{p})}{(k + b_{1})(k + b_{2}) ⋯ (k + b_{q})(k + 1)}x.

Proposition 4.5. A little simplifying LSMI < a! > and stating in standard notations, we have

∑_{k=1}^{n} {(k − 1)^{2}k! + (2k − 1)(k + 1!)ℌ(1, k + 2)} = n^{2}(n + 1)!ℌ(1, n + 2),

where

ℌ(α, β) = _{2}F_{0} [ \begin{matrix}α & β \\ − \end{matrix} ; 1] .

Proof. Considering definition of hypergeometric functions we have ℌ(1, n + 1) = (n + 1)ℌ(1, n + 2), which implies ∑_{a=1}^{n} a! = ℌ(1, 2) − (n + 1)!ℌ(1, n + 2) = 𝔓(n), say. This gives Σ(n) = n^{2}𝔓(n) and in similar way it yields that L_{k} = (k − 1)^{2}k! + (2k − 1) ((k + 1!)ℌ(1, k + 2) − ℌ(1, 2)). Thus, we obtain

∑_{k=1}^{n} {(k − 1)^{2}k! + (2k − 1) ((k + 1!)ℌ(1, k + 2) − ℌ(1, 2))} = n^{2}(n + 1)!ℌ(1, n + 2) − n^{2}ℌ(1, 2),

and a easy simplifying this, implies the result. □

Remark 4.6. Three last propositions are examples of the array A_{abc} = f(a), for some given function f. In this case, L−summing method takes the following form

∑_{k=1}^{n} {(2k − 1)𝔉(k) + (k − 1)^{2}f(k)} = n^{2}𝔉(n),

where 𝔉(n) = ∑_{a=1}^{n} f(a).

5. Further generalizations of the L−summing method and some comments

5.1. The L−Summing method in ℝ^{t}. Consider a t−dimensional array A_{x_{1}x_{2}···x_{t}} and let Σ(n) = ∑ A_{x_{1}x_{2}···x_{t}} with 1 ≤ x_{1}, x_{2}, ⋯, x_{t} ≤ n. The L−Summing method in ℝ^{t} is the rearrangement Σ(n) = ∑ L_{k}, where L_{k} = ∑_{m=1}^{t} {(−1)^{m−1}Σ_{t−m}} and

Σ_{t−m} = ∑_{1≤i_{1}<i_{2}<···<i_{m}≤t} {∑′A_{x_{i_{1}i_{2}···im}} } ,

where in the inner summation ∑′ is over x_{j} ∈ {x_{i_{1}}, ⋯, x_{i_{m}}}^{C} = {x_{1}, x_{2}, ⋯, x_{t}} − {x_{i_{1}}, ⋯, x_{i_{m}}} with 1 ≤ x_{j} ≤ k, and the index x_{i_{1}i_{2}···i_{m}} denotes x_{1}x_{2} ⋯ x_{t} with x_{i_{1}} = x_{i_{2}} = ⋯ = x_{i_{m}} = k. One can apply this generalized version to get more general form of relations obtained in previous sections. For example, considering the array A_{x_{1}x_{2}···x_{t}} = (x_{1}x_{2} ⋯ x_{t})^{−s} with s ∈ ℂ, yields

∑_{k=1}^{n} \{ ∑_{m=1}^{t−1}(−1)^{m−1}\binom{t}{m}k^{−ms}ζ_{k}(s)^{t−m}\} = ζ_{n}(s)^{t} + (−1)^{t}ζ_{n}(ts).

5.2. L−summing method on manifolds. As in rising of this paper, the base of the L−summing method is ordinary multiplications table. Above generalization of the L−Summing method in ℝ^{t} is based on generalized multiplication tables [4]. But, ℝ^{t} is a very special t-dimensional manifold, and if we replace it with Γ, an l−dimensional manifold with l ≤ t, then we can define generalized multiplication table on Γ by considering lattice points on it (which of course isn’t easy problem). Let

L_{Γ}(n) = {(a_{1}, a_{2}, ⋯, a_{t}) ∈ Γ ∩ ℕ^{t} : 1 ≤ a_{1}, a_{2}, ⋯, a_{t} ≤ n},

and f : ℝ^{k} ⟶ ℂ is a function. If O_{Γ} is a collection of k − 1 dimension orthogonal manifolds, in which L_{Γ}(n) = ∪_{Λ∈O_{Γ}}L_{Λ}(n) and L_{Λ_{i}}(n) ∩ L_{Λ_{j}}(n) = ϕ for distinct Λ_{i}, Λ_{j} ∈ O_{Γ}, then we can formulate L−summing method as follows,

∑_{X∈L_{Γ}(n)} f(X) = ∑_{Λ∈O_{Γ}} { ∑_{X∈L_{Λ}(n)} f(X)} .

Here L−summing elements are ∑_{X∈L_{Λ}(n)} f(X). This may be useful when one apply it on some special manifolds.

5.3. Stronger form of L−summing method. One can state the method of L−summing ∑ L_{k} = Σ(n) in the following stronger form

L_{n} = Σ(n) − Σ(n − 1).

Specially, this will be useful for those arrays with Σ(n) computable explicitly and L_{k} maybe note. For example, applying this note on the array A_{x_{1}x_{2}···x_{t}} = (x_{1}x_{2} ⋯ x_{t})^{−s} in ℝ^{t} with s ∈ ℂ, implies

∑_{m=1}^{t−1}(−1)^{m−1}\binom{t}{m}n^{−ms}ζ_{n}(s)^{t−m} = ζ_{n}(s)^{t} + (−1)^{t}ζ_{n}(ts) − ζ_{n−1}(s)^{t} − (−1)^{t}ζ_{n−1}(ts).

References

[1] M. Abramowitz and I.A. Stegun, HANDBOOK OF MATHEMATICAL FUNCTIONS: with Formulas, Graphs, and Mthematical Tables, Dover Publications, 1972.

    Available at: http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP

[2] Jacek Gilewicz (Reviewer), Review of the paper [3] (in this reference list), Zentralblatt MATH Database, European Mathematical Society, FIZ Karlsruhe & Springer-Verlag, 2007.

[3] M. Hassani, Identities by L-summing Method, Int. J. Math. Comp. Sci., Vol. 1 (2006) no. 2, 165-172.

[4] M. Hassani, Lattice Points and Multiplication Tables, Int. J. Contemp. Math. Sci., Vol. 1 (2006) no. 1, 1-2.

[5] Aleksandar Ivic, The Riemann Zeta Function, John Wiley & sons, 1985.

[6] M. Petkovšek, H.S. Wilf and D. Zeilberger, A = B, A. K. Peters, 1996.

    Available at: http://www.cis.upenn.edu/ wilf/AeqB.html

Appendix. Maple Program of 3-dimension L−Summing Method for the array A_{abc} = \frac{1}{abc}

restart:
A[abc]:=1/(a*b*c);
S21:=sum(sum(eval(A[abc],a=k),b=1..k),c=1..k):
S22:=sum(sum(eval(A[abc],b=k),a=1..k),c=1..k):
S23:=sum(sum(eval(A[abc],c=k),a=1..k),b=1..k):
S2:=S21+S22+S23:
S11:=sum(eval(eval(A[abc],a=k),b=k),c=1..k):
S12:=sum(eval(eval(A[abc],a=k),c=k),b=1..k):
S13:=sum(eval(eval(A[abc],b=k),c=k),a=1..k):
S1:=S11+S12+S13:
S0:=eval(eval(eval(A[abc],a=k),b=k),c=k):
L[k]:=simplify(S2-S1+S0):
ST(A):=(simplify(sum(sum(sum(A[abc],a=1..n),b=1..n),c=1..n))):
Sum(L[k],k=1..n)=ST(A);

A_{abc} := \frac{1}{abc}

∑_{k=1}^{n} \frac{3 (Ψ (k + 1))^{2} k^{2} + 6 Ψ (k + 1) k^{2}γ + 3 γ^{2}k^{2} − 3 Ψ (k + 1) k − 3 γ k + 1}{k^{3}} = (Ψ (n + 1) + γ)^{3}

Mehdi Hassani, Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

   E-mail address: mmhassany@member.ams.org

Zahra Jafari, Department of Information Technology, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

   E-mail address: z.jafari@iasbs.ac.ir, zahra7jfr@yahoo.com