Evaluation of weighted Fibonacci sums of a certain type

TL;DR

This paper derives a formula for evaluating weighted sums involving generalized Fibonacci numbers, providing explicit examples and expanding the understanding of Fibonacci sum evaluations.

Contribution

It introduces a new formula for weighted generalized Fibonacci sums of a specific type, with several explicit evaluations as examples.

Findings

Derived a general formula for weighted Fibonacci sums

Provided explicit evaluations for specific cases

Enhanced methods for Fibonacci sum calculations

Abstract

We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.

Equations

$$S_k^n(w,r)=\sum _{j=0}^k{w}^j{j}^r{G}_j{}^n,$$

$$\sum _{j=a}^b{f}_j\equiv -\sum _{j=b+1}^{a-1}{f}_j.$$

$$\sum _{j=a+1}^{a-1}{f}_j=-f_a\text{mbox}and\sum _{j=a}^{a-1}{f}_j=0.$$

$$D=w\frac{\mathrm{d}}{\mathrm{d}w},$$

$$\begin{array}{rl}\sum _{j=1}^k{j}^5{G}_j& =2671G_0+4322G_1+(k^5-5k^4+50k^3-310k^2+1285k-2671)G_k\\ & +(k^5-10k^4+80k^3-500k^2+2080k-4322)G_{k+1},\end{array}$$

$$\sum _{j=1}^k(-1{)}^{j}j{G}_j=3G_0-2G_1-(-1{)}^k(k+3)G_k+(-1{)}^k(k+2)G_{k+1},$$

$$\sum _{j=1}^k\frac{j{G}_j}{2^j}=6G_0+10G_1-(k+6)\frac{G_k}{2^k}-(k+5)\frac{G_{k+1}}{2^{k-1}},$$

$$\sum _{j=1}^k{2}^{j}j{G}_j=\frac{2}{5}{G}_1+\frac{2^{k+2}}{5}k{G}_k+\frac{2^{k+1}}{5}(k-1)G_{k+1},$$

$$\sum _{j=1}^k(-1{)}^j{G}_{2j}=\frac{3}{5}{G}_0-\frac{1}{5}{G}_1+(-1{)}^k\frac{2}{5}{G}_{2k}+(-1{)}^k\frac{1}{5}{G}_{2k+1},$$

$$\sum _{j=1}^k(-1{)}^{j-1}{G}_{2j-1}=-\frac{1}{5}{G}_0+\frac{2}{5}{G}_1+(-1{)}^k\frac{1}{5}{G}_{2k}-(-1{)}^k\frac{2}{5}{G}_{2k+1}.$$

$$\sum _{j=0}^{\infty }\frac{G_j^2}{2^{4j}}=\frac{3552}{3553}{G}_0^2+\frac{224}{3553}{G}_1^2+\frac{16}{3553}{G}_2^2,$$

$$\begin{array}{rl}\sum _{j=0}^k(-1{)}^j{G}_{2j}^3& =\frac{23}{25}{G}_0^3-\frac{9}{50}{G}_1^3-\frac{13}{50}{G}_2^3+\frac{3}{50}{G}_3^3-\frac{3}{50}(-1{)}^k{G}_{2k+1}^3\\ & +\frac{37}{50}(-1{)}^k{G}_{2k+2}^3+\frac{9}{50}(-1{)}^k{G}_{2k+3}^3-\frac{2}{25}(-1{)}^k{G}_{2k+4}^3.\end{array}$$

$$S_k(w,0)=\sum _{j=0}^k{w}^j{G}_j=\frac{-(w-1)G_0+w{G}_1-w^{k+2}{G}_k-w^{k+1}{G}_{k+1}}{1-w-w^2}.$$

$$S_k(1,0)=\sum _{j=0}^k{G}_j=G_{k+2}-G_1.$$

$$\begin{array}{rl}\sum _{j=0}^k{w}^j{G}_j& =\sum _{j=0}^k{w}^j{G}_{j-1}+\sum _{j=0}^k{w}^j{G}_{j-2}\\ & =w\sum _{j=-1}^{k-1}{w}^j{G}_j+w^2\sum _{j=-2}^{k-2}{w}^j{G}_j\end{array}$$

$$\begin{array}{rl}& =w\left(\frac{G_{-1}}{w}+\sum _{j=0}^k{w}^j{G}_j-w^k{G}_k\right)\\ & +w^2\left(\frac{G_{-2}}{w^2}+\frac{G_{-1}}{w}+\sum _{j=0}^k{w}^j{G}_j-w^{k-1}{G}_{k-1}-w^k{G}_k\right),\end{array}$$

$$\begin{array}{rl}{S}_k(w,0)& =G_{-1}+w{S}_k(w,0)-w^{k+1}{G}_k\\ & +G_{-2}+w{G}_{-1}+w^2{S}_k(w,0)\\ & -w^{k+1}{G}_{k-1}-w^{k+2}{G}_k,\end{array}$$

$$S_{\infty }(w,0)=\sum _{j=0}^{\infty }{w}^j{G}_j=\frac{(1-w)G_0+w{G}_1}{1-w-w^2}.$$

$$A(w;m)=D^m\frac{1}{1-w-w^2}=D^{m}A(w;0),$$

$$(1-w-w^2)A(w;m)={\delta }_{m0}+w\sum _{j=0}^{m-1}\left(\genfrac{}{}{0ex}{}{m}{j}\right)(2^{m-j}w+1)A(w;j),m=0,1,2,\dots .$$

$$\begin{array}{rl}{S}_k(w,r)=\sum _{j=0}^k{w}^j{j}^r{G}_j& =-G_0\sum _{m=0}^r\left(\genfrac{}{}{0ex}{}{r}{m}\right)(w-{\delta }_{rm})A(w;m)+w{G}_1\sum _{m=0}^r\left(\genfrac{}{}{0ex}{}{r}{m}\right)A(w;m)\\ & -w^{k+2}{G}_k\sum _{m=0}^r\left(\genfrac{}{}{0ex}{}{r}{m}\right)(k+2{)}^{m}A(w;r-m)\\ & -w^{k+1}{G}_{k+1}\sum _{m=0}^r\left(\genfrac{}{}{0ex}{}{r}{m}\right)(k+1{)}^{m}A(w;r-m),\end{array}$$

$$(1-w-w^2)A(w;p)={\delta }_{p0}+w\sum _{j=0}^{p-1}\left(\genfrac{}{}{0ex}{}{p}{j}\right)(2^{p-j}w+1)A(w;j).$$

$$\sum _{j=0}^k\frac{G_j}{2^j}=2G_0+2G_1-\frac{G_k}{2^k}-\frac{G_{k+1}}{2^{k-1}},$$

$$\sum _{j=0}^{\infty }\frac{G_j}{2^j}=2G_0+2G_1.$$

$$\sum _{j=0}^k\frac{j{G}_j}{2^j}=6G_0+10G_1-(k+6)\frac{G_k}{2^k}-(k+5)\frac{G_{k+1}}{2^{k-1}},$$

$$\sum _{j=0}^{\infty }\frac{j{G}_j}{2^j}=6G_0+10G_1.$$

$$\begin{array}{rl}\sum _{j=0}^k\frac{j^2{G}_j}{2^j}& =58G_0+94G_1-(k^2+12k+58)\frac{G_k}{2^k}\\ & -(k^2+10k+47)\frac{G_{k+1}}{2^{k-1}},\end{array}$$

$$\sum _{j=0}^{\infty }\frac{j^2{G}_j}{2^j}=58G_0+94G_1.$$

$$\sum _{j=0}^{2k}{f}_j=\sum _{j=0}^k{f}_{2j}+\sum _{j=1}^k{f}_{2j-1}$$

$$\sum _{j=0}^{2k}{i}^j{j}^r{G}_j=2^r\sum _{j=0}^k(-1{)}^j{j}^r{G}_{2j}+i\sum _{j=1}^k(-1{)}^{j-1}(2j-1{)}^r{G}_{2j-1}.$$

Full text

1 Introduction

The generalized Fibonacci numbers $G_{i}$, $i\geq 0$, are defined through the second order recurrence relation $G_{i+1}=G_{i}+G_{i-1}$, with arbitrary seeds $G_{0}$ and $G_{1}$. In particular, when $G_{0}=0$ and $G_{1}=1$, we have the Fibonacci numbers, denoted $F_{i}$, and when $G_{0}=2$ and $G_{1}=1$, we have the Lucas numbers, $L_{i}$. Extension to negative index is provided through $G_{-i}=(-1)^{i}(F_{i+1}G_{0}-F_{i}G_{1})$, Vajda [12], Formula (9).

In this paper we will derive a formula (Theorem 3.2) for the evaluation of weighted generalized Fibonacci sums of the type

[TABLE]

where $w$ is a real or complex parameter and $r$, $k$ and $n$ are nonnegative integers. For brevity, $S_{k}^{1}(w,r)$ will be denoted $S_{k}(w,r)$ and $S_{k}^{1}(1,r)$ will often be denoted simply by $S_{k}(r)$. Similarly, we will write $\overline{S}_{k}(r)$ for $S_{k}^{1}(-1,r)$. The evaluation of $S_{k}(r)=\sum_{j=0}^{k}{j^{r}G_{j}}$ has received some attention in the literature, see for example references [10, 9, 2, 5, 11].

Since $S_{k}^{n}(w,0)$ in (1.1) is to give $\sum_{j=0}^{k}{w^{j}G_{j}{}^{n}}$, we shall adopt the convention that $j^{0}=1$ for all $j$, including $j=0$.

For evaluating sums with $a>b$ we shall use

[TABLE]

In particular

[TABLE]

Following Gauthier [4], we introduce the differential operator

[TABLE]

and note the following properties:

$D^{0}u(w)=u(w)$. 2. 2.

$D(au(w)+bv(w))=aD(u(w))+bD(v(w))$, for complex numbers $a$ and $b$. 3. 3.

$D^{p}\left({u(w)v(w)}\right)=\sum_{r=0}^{p}{\binom{p}{r}D^{r}u(w)D^{p-r}v(w)}$ for $p$ times differentiable functions $u(w)$ and $v(w)$. 4. 4.

If $u(w)=\sum_{r=0}^{n}{a_{r}w^{r}}$, where the coefficients $a_{r}$ are real or complex numbers and $n$ is an integer, then, $D^{p}u(w)=a_{0}\delta_{p0}+\sum_{r=1}^{n}{r^{p}a_{r}w^{r}}$, where $\delta_{ij}$ is the usual Kronecker delta. 5. 5.

$D^{p}S_{k}^{n}(w,0)=S_{k}^{n}(w,p)$ .

The strength of our analysis lies in the properties 3, 4 and 5 which are easily verified by induction on $p$.

Here is a couple of anticipated evaluations to whet the reader’s appetite for reading on.

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2 Evaluation of $S_{k}(w,r)=\sum_{j=0}^{k}{w^{j}j^{r}G_{j}}$

Lemma 1**.**

If $k$ is a positive integer and $w$ is a parameter, real or complex, then

[TABLE]

In particular, we have

[TABLE]

Proof.

Multiply through the recurrence relation $G_{j}=G_{j-1}+G_{j-2}$ by $w^{j}$ and sum over $j$, obtaining

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[TABLE]

so that we have

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and the result follows.

∎

Note that in the identity of Lemma 1, if we let $k$ approach infinity and require $w^{k}G_{k}$ to vanish as $k$ approaches infinity, then we obtain the generating function for $G_{i}$, namely,

[TABLE]

Lemma 2**.**

The rational function

[TABLE]

satisfies the following recursion relation

[TABLE]

Proof.

Write $(1-w-w^{2})A(w;0)=1$ and apply $D^{m}$ to both sides. ∎

Theorem 2.1**.**

If $k$ and $r$ are nonnegative integers and $w$ a real or complex parameter, then

[TABLE]

where the $A(w;p)$, $p=0,1,2,\ldots,r$, are given recursively by

[TABLE]

Proof.

Apply $D^{r}$ to both sides of the identity of Lemma 1 and make use of Lemma 2 and the properties 3, 4 and 5 of the $D$ operator. ∎

Numerous interesting summation formulas can be obtained directly from Theorem 2.1. By setting $w=1/2$, $r=0,1,2$, respectively, we already have some examples:

[TABLE]

corresponding to Vajda [12], Formula (37), proved by induction, which in the limit as $k\to\infty$ gives

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Note that the identity (2.5) subsumes Formulas (A3.52) and (A3.53) of Dunlap [3].

If $i=\sqrt{-1}$ is the imaginary unit, then using $f_{j}=i^{j}j^{r}G_{j}$ in the summation identity

[TABLE]

allows us to write

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Thus, setting $w=i$ in Theorem 2.1 and making use of identity (2.8), we have the following results for $r=0,1$.

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Identities (2.9) and (2.10) are the alternating counterparts of Formulas (34) and (35) of Vajda [12].

Corollary 2.2**.**

If $k$ is a nonnegative integer and $w$ is a real or complex parameter, then

[TABLE]

In particular, we have

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and

[TABLE]

Corollary 2.3**.**

If $r$ is a nonnegative integer and $w$ is a real or complex parameter such that $w^{k}G_{k}\to 0$ as $k\to\infty$ for all positive integers $k$, then

[TABLE]

Thus, Corollary 2.3 provides the evaluation of generating functions for $j^{r}G_{j}$. A strategy for using probability to evaluate $S_{\infty}(1/2,r)$ was discussed in reference [1]. Dropping the last two terms in Corollary 2.2, we have

[TABLE]

a result that was also obtained and whose convergence was exhaustively discussed by Glaister [6, 7] for the Fibonacci case, (G=F).

Corollary 2.4**.**

If $k$ and $r$ are nonnegative integers, then

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where the $A(m)$ satisfy the recurrence relation

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Here is a little list from Corollary 2.4.

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Corollary 2.5**.**

If $k$ and $r$ are nonnegative integers, then

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where the $\overline{A}(p)$ are given recursively by

[TABLE]

Here are a few evaluations.

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[TABLE]

3 Evaluation of $S_{k}^{n}(w,r)=\sum_{j=0}^{k}{w^{j}j^{r}G_{j}{}^{n}}$

Lemma 3**.**

If $k$ and $n$ are nonnegative integers and $w$ is a parameter, real or complex, then

[TABLE]

where $\lceil u\rceil$ is the smallest integer greater than $u$ and the Fibonomial coefficients are defined by

[TABLE]

Proof.

Multiply through the following identity [8, section 1.2.8, Exercise 30] by $w^{j}$

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and sum over $j$, obtaining

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which after shifting the index $j$ becomes

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Since

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we have

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Thus,

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Clearing brackets, multiplying through by $w^{n+1}$ and making $S_{k}^{n}(w,0)$ the subject, the lemma is proved. ∎

In particular, we have

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and

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Interesting evaluations can already be obtained from (3.1) and (3.2). For example, at $w=1$, we have

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and

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while $w=1/2$ gives

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and

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At $w=2$, we have

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and

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At $w=1/16$, we have

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giving

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Evaluation at $w=\sqrt{-1}=i$, $n=2,3$ gives

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[TABLE]

[TABLE]

and

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Corollary 3.1** (Generating function for $G_{j}{}^{n}$).**

If $n$ is a nonnegative integer and $w$ is a parameter, real or complex, such that $w^{k}G_{k}{}^{n}$ vanishes as $k$ approaches infinity, then

[TABLE]

In particular, the generating functions of $G_{j}{}^{2}$ and $G_{j}{}^{3}$ are, respectively,

[TABLE]

and

[TABLE]

Lemma 4**.**

The rational function

[TABLE]

satisfies the following recurrence relation

[TABLE]

Proof.

Write

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Apply $D^{m}$ to both sides to obtain

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from which the result follows.

∎

Theorem 3.2**.**

If $k$, $r$ and $n$ are nonnegative integers and $w$ is a real or complex parameter, then

[TABLE]

where the functions $A_{n}(w,m)$ are given recursively by

[TABLE]

for $m=0,1,2,\ldots,r$.

Proof.

Apply $D^{r}$ to both sides of the identity of Lemma 3 and make use of Lemma 4 and the properties 3, 4 and 5 of the $D$ operator. ∎