I. Introduction

In this Note we will analyze the finite sums

(1)   

for p = 0, 1, 2, 3,.... and real x (see Section IV for extensions).
These can be considered as the partial sums of the power series whose generic term is k^px^k. Using elementary convergence criteria it can be easily shown that the series converges for any real or complex x with modulus smaller than 1. The finite partial sums, however, are well defined for any value of x.

For p = 0, expression (1) this reduces to the truncated geometric series for which

(2)   

The singularity at x = 1 is only apparent since the nominator in Eq.(2) also has a root at x = 1.
It is instructive to notice that

(3)   ,

where the δ symbol is equal to 1 for p = 0 and to 0 otherwise.
This shows that p = 0 is a special case and anticipates the fact that some of the formulae have a different form for p = 0 and p > 0.

In the next Section we will prove that for p > 0 the sums of Eq.(1) can be cast into the form

(4)   ,

where P_p(x) is a polynomial in x of order (p-1) and Q_p(x,n) is a polynomial of order p in both x and n. Since the finite sum of Eq.(1) is well behaved even for x = 1, the term in the square brackets in Eq.(4) must have at x = 1 a root of multiplicity (p+1).

When x < 1, the limit of S_p,n(x) for n growing to infinity exists and, according to Eq.(4), the sum of the resulting power series is

(5)   .

This means that, under the same convergence conditions,

(6)   

which, apart from a pleasing analogy with Eq.(5), confers a more direct meaning to the bi-polynomials Q_p(x,n).

By definition, S_p,0(x) = 0 for any p > 0 and any x. It follows that

(7)   ,

a relation which can be used as a check.

The purpose of this Note is to prove Eq.(4), derive formulas for the polynomials P_p(x) and Q_p(x,n), analyze some of their properties, and tabulate them for a few values of p.

II. Proof of the explicit summation formula

An elementary inspection of Eq.(1) shows the validity of the following recurrence, valid for any p and any finite n

(8)   .

For p = 0 this gives

(9)   ,

thus confirming the validity of Eq.(4) for p = 1, with

(10)   .

We shall now prove that the validity of Eq.(4) for some p > 0 implies its validity for (p+1).
From Eq.(6) we have

(11)   

where

(12)   

and

(13)   .

From Eq.(10) it is clear that when P_p(x) is a polynomial of degree (p-1) than P_p+1(x) is indeed a polynomial of degree p. Likewise, Eq.(11) shows that if Q_p(x,n) is a polynomial of degree p in x and n, then Q_p+1(x,n) is a polynomial of degree (p+1) in both x and n. QED.

III. Determination of the polynomial coefficients

a) The infinite series polynomials P_p(x)

Writing

(14)   ,

equation (12) gives the following recurrence for the coefficients

(15)   .

The coefficients turn out to possess the following symmetry property

(16)   ,

which can be easily proved by recurrence using Eq.(15) and the fact that it is trivial for p = 1.
 

Consulting the Table and Eq.(5), the explicit expressions for the sum of the infinite series can be written as

(17)   

etc. Substituting for x specific values with |x| < 1, this leads to a number of particular infinite sums. In particular,

(18)    for p = 0, 1, 2, 3, 4, 5, ..., respectively (OEIS A076726).
For x = -1/2 one obtains

(19)   , again for p = 0, 1, 2, 3, 4, 5, ..., respectively.

For larger values of p the absolute values of this series become very large, while their signs continue to be quite unpredictable. For example, for p = 18 the result is about 2431084.21 while for p = 19 it is -6262402.20... The question of whether there is a finite or infinite number of sign changes in the sequence {S_p(-1/2)} looks like an unsolved problem of considerable complexity.
The denominators in (19) are integer powers of 3 and form the sequence OEIS A131137.

b) The polynomials Q_p(x,n)

Since Q_p(x,n) is a polynomial of degree p in both x and n, it can be written as

(20)   ,

where C_p(m,ν) are numeric coefficients forming a square matrix of order (p+1) which we will call just C-matrix. Substituting Eq.(20) into Eq.(13) and comparing the coefficients, one obtains the following recurrence formula for the elements of the C-matrices:

(21)   ,

which, combined with the start-up coefficients given by Eq.(10),

(22)   ,

and with the rule that C_p(m,ν) = 0 whenever m and/or ν is either negative or greater than p, permits the determination of the C-matrices to any desired order.

In the C-matrices shown on the right the row index is m (i.e., the power of x) and the column index is ν (i.e., the power of n). In both cases, the numbering starts from 0. Thus, for example, Q₂(x,n) can be written either as

(23a)   

or as

(23b)   .

Consequently, using the second form

(24)   .

Like the c_p,k coefficients of the P_p(x) polynomials (Eq.16), the C_p(m,ν) satisfy a number of relations which are also easy to prove by recursion directly from Eqs.(21-22). Thus the sum of any column except the first one is null, the first row evaluates to (1+n)^p, the last row evaluates to (-xn)^p and, for m < p,

(26)   .

IV. Alternative recurrences

It is relatively easy to find a direct recurrence for S_p,n(x).
Starting directly from the definition, one can proceed as follows:

(27)   ,

where C(n,m) is used to denote the binomial coefficients. This amounts to a direct recurrence for S_p,n(x)

(28)   .

Assuming |x| < 1 in order to guarantee convergence, and taking the limit of Eq.(28) for large n, it simplifies to a direct recurrence for the infinite series S_p(x) which makes evaluation of sequences like that in (19) relatively straightforward.

(29)   .

The direct recurrences are often numerically more comfortable to use than tabulated polynomial expressions.

It is also clear that assuming the functional form of Eq.(4), one could easily work out direct recurrences for the polynomial values P_p(x) and Q_p(x,n), without passing through their coefficients, but that is not the goal of this essay.

V. Special cases of finite sums

Using Eq.(4) and the known expressions for P_p(x) and Q_p(x,n) one can now easily derive a number of notable finite-sum identities. For completeness sake, we list here also the results for p = 0 as given by Eq.(2).

a) Special cases for x = -1 (respectively, OEIS A059841, A130472, A089594, A232599)

   .

b) Special cases for x = 2 (respectively, OEIS A232733, A036799, A036800, A036827)

   .

c) Special cases for x = -2 (respectively, OEIS A077925, A232600, A232601, A232602)

   .

d) Special cases for x = 1/2 (multiplied by 2 ⁿ, respectively, OEIS A000225, A000295, A047520, A213575)

   .

e) Special cases for x = -1/2 (multiplied by 2 ⁿ, respectively, OEIS A001045, A053088, A232603, A232604)

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