# Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences

@article{Stnic2000GeneratingFW, title={Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences}, author={Pantelimon Stănică}, journal={arXiv: Combinatorics}, year={2000} }

In this paper we find closed form for the generating function of powers of any non-degenerate second-order recurrence sequence, completing a study begun by Carlitz and Riordan in 1962. Moreover, we generalize a theorem of Horadam on partial sums involving such sequences. Also, we find closed forms for weighted (by binomial coefficients) partial sums of powers of any non-degenerate second-order recurrence sequences. As corollaries we give some known and seemingly unknown identities and derive…

## 13 Citations

### Several Generating Functions for Second-Order Recurrence Sequences

- Mathematics
- 2009

This paper considers exponential and other types of generating functions for powers of second-order recurrence sequences and an extensive table of generating Functions for such sequences is provided.

### Some Weighted Sums of Products of Lucas Sequences

- MathematicsIntegers
- 2013

The weighted sums of products of Lucas sequences of the form n ∑ k=0 ( n k ) rmksm(tn+k) are considered and it is shown that these sums could be expressed via terms of the Lucas sequences.

### Weighted sums of some second-order sequences.

- Mathematics
- 2018

We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where…

### On sums related to the numerator of generating functions for the kth power of Fibonacci numbers on sums related to the numerator of generating functions for the kth power of Fibonacci numbers

- Mathematics
- 2010

AbstractNew results about some sums sn(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J.…

### SUMS OF PRODUCTS OF THE TERMS OF THE GENERALIZED LUCAS SEQUENCE { V kn }

- Mathematics
- 2011

In this study we consider the generalized Lucas sequence {Vn} with indices in arithmetic progression. We also compute the sums of products of the terms of the Lucas sequence {Vkn} for positive odd…

### A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters

- Mathematics
- 2011

In this note, we give a generalization of a formula for the generating function of powers of Horadam’s sequence by adding two parameters. Thus we obtain a generalization of a formula of Mansour.

### More on Second-Order Linear Recurrent Sequences

- Mathematics
- 2020

The sequence (xn)n≥0 = (xn(a, b;p, q))n≥0 defined by
$$\displaystyle x_{n+2} = px_{n+1} + qx_n, \quad x_0 = a, x_1 = b, $$
with a, b, p, and q arbitrary complex numbers is called a Horadam…

### Partial sums and generating functions of products of Horadam numbers with indices in arithmetic progression

- Mathematics
- 2019

The sums $\sum_{j = 0}^k {w_{rj + s} u_{mj + n} z^j }$, $\sum_{j = 0}^k {w_{rj + s} v_{mj + n} z^j }$ and $\sum_{j = 0}^k {w_{rj + s} w_{mj + n} z^j }$ are evaluated; where $r$, $s$, $k$, $m$ and $n$…

### THE FIBONACCI QUARTERLY

- Mathematics
- 2010

. Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result holds for other positive linear recurrence sequences. These…

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