New identities for some symmetric polynomials, and a higher order analogue of the Fibonacci and Lucas numbers

TL;DR

This paper introduces new identities for symmetric polynomials and applies them to derive formulas for higher order analogues of Fibonacci and Lucas numbers, expanding understanding of these classical sequences.

Contribution

It presents novel identities for symmetric polynomials and uses them to develop higher order Fibonacci and Lucas number formulas, a new approach in this area.

Findings

New identities for symmetric polynomials

Formulas for higher order Fibonacci numbers

Formulas for higher order Lucas numbers

Abstract

We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.

Tables

Table 1. Table 1: c n , k subscript 𝑐 𝑛 𝑘 c_{n,k}

$n\setminus k$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$0$	$1$
$1$	$1$
$2$	$1$	$1$
$3$	$1$	$2$
$4$	$1$	$3$	$2$
$5$	$1$	$4$	$5$
$6$	$1$	$5$	$9$	$5$
$7$	$1$	$6$	$14$	$14$
$8$	$1$	$7$	$20$	$28$	$14$
$9$	$1$	$8$	$27$	$48$	$42$
$10$	$1$	$9$	$35$	$75$	$90$	$42$
$11$	$1$	$10$	$44$	$110$	$165$	$132$
$12$	$1$	$11$	$54$	$154$	$275$	$297$	$132$
$13$	$1$	$12$	$65$	$208$	$429$	$572$	$429$
$14$	$1$	$13$	$77$	$273$	$637$	$1001$	$1001$	$429$
$15$	$1$	$14$	$90$	$350$	$910$	$1638$	$2002$	$1430$
$16$	$1$	$15$	$104$	$440$	$1260$	$2548$	$3640$	$3432$	$1430$
$17$	$1$	$16$	$119$	$544$	$1700$	$3808$	$6188$	$7072$	$4862$
$18$	$1$	$17$	$135$	$663$	$2244$	$5508$	$9996$	$13260$	$11934$	$4862$
$19$	$1$	$18$	$152$	$798$	$2907$	$7752$	$15504$	$23256$	$25194$	$16796$
$20$	$1$	$19$	$170$	$950$	$3705$	$10659$	$23256$	$38760$	$48450$	$41990$	$16796$
$21$	$1$	$20$	$189$	$1120$	$4655$	$14364$	$33915$	$62016$	$87210$	$90440$	$58786$

Table 2. Table 2: F n ( r ) superscript subscript 𝐹 𝑛 𝑟 F_{n}^{(r)}

$r\setminus n$	1	2	3	4	5	6	7	8	9	10	11	12
1	1	1	1	1	1	1	1	1	1	1	1	1
2	1	1	2	3	5	8	13	21	34	55	89	144
3	1	1	3	4	9	14	28	47	89	155	286	507
4	1	1	4	5	14	20	48	75	165	274	571	988
5	1	1	5	6	20	27	75	110	275	429	1001	1637
6	1	1	6	7	27	35	110	154	429	637	1638	2548
7	1	1	7	8	35	44	154	208	637	910	2548	3808
8	1	1	8	9	44	54	208	273	910	1260	3808	5508
9	1	1	9	10	54	65	273	350	1260	1700	5508	7752
10	1	1	10	11	65	77	350	440	1700	2244	7752	10659
11	1	1	11	12	77	90	440	544	2244	2907	10659	14364
12	1	1	12	13	90	104	544	663	2907	3705	14364	19019
13	1	1	13	14	104	119	663	798	3705	4655	19019	24794
14	1	1	14	15	119	135	798	950	4655	5775	24794	31878
15	1	1	15	16	135	152	950	1120	5775	7084	31878	40480
16	1	1	16	17	152	170	1120	1309	7084	8602	40480	50830

Equations

$$e_n^{(r)}$$

$$h_n^{(r)}$$

$$p_n^{(r)}$$

$$(\mathbf{z}+{\mathbf{z}}^{-1})$$

$$(\mathbf{z},{\mathbf{z}}^{-1})$$

$$f_n^{(r)}(\mathbf{z}+{\mathbf{z}}^{-1})$$

$$f_n^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1})$$

$$\sum _{k=\max \left(\lfloor \frac{m}{2}\rfloor -r,0\right)}^{\lfloor \frac{m}{2}\rfloor }{c}_{m-r-1,k}{e}_{m-2k}^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1})=\{\begin{array}{ll}{e}_m^{(r)}(\mathbf{z}+{\mathbf{z}}^{-1})& (m=0,1,\dots ,r)\\ 0& (\mathrm{others})\end{array},$$

$$c_{n,k}:=\left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n}{k-1}\right),\left(\genfrac{}{}{0ex}{}{n}{k}\right):=\{\begin{array}{ll}\frac{n(n-1)\cdots (n-k+1)}{k!}& (k\ne 0)\\ 1& (k=0)\end{array}.$$

$$h_m^{(r)}(\mathbf{z}+{\mathbf{z}}^{-1})=\sum _{k=0}^{\lfloor \frac{m}{2}\rfloor }{c}_{m+r-1,k}{h}_{m-2k}^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1}).$$

$$p_m^{(r)}(\mathbf{z}+{\mathbf{z}}^{-1})=\frac{1}{2}\sum _{k=0}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)p_{|m-2k|}^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1}).$$

$$e_n^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1})$$

$$h_n^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1})$$

$$p_n^{(2r)}(\mathbf{z},{\mathbf{z}}^{-1})$$

$$F_{n+1}^{(r)}$$

$$L_n^{(r)}$$

$${\mathbf{\zeta }}^{\pm \iota }:=\left(e^{\pm \frac{2\pi \sqrt{-1}r}{2r+1}},\dots ,e^{\pm \frac{2\pi \sqrt{-1}}{2r+1}}\right).$$

$$F_{n+1}^{(1)}=L_n^{(1)}=1,(n\ge 0)$$

$$F_{n+1}^{(2)}$$

$$L_n^{(2)}$$

$$F_n^{(r)}$$

$$=\sum _{k=0}^{\lfloor \frac{n-1}{2r+1}\rfloor }(-1{)}^k{c}_{n+r-2,\lfloor \frac{n-1-(2r+1)k}{2}\rfloor },$$

$$L_n^{(r)}$$

$$=\{\begin{array}{ll}-2^{2m-1}+\frac{2r+1}{2}{\sum }_{k=-\lfloor \frac{m}{2r+1}\rfloor }^{\lfloor \frac{m}{2r+1}\rfloor }\left(\genfrac{}{}{0ex}{}{2m}{m-(2r+1)k}\right)& (n=2m)\\ 4^m-\frac{2r+1}{2}{\sum }_{k=-\lfloor \frac{m+r+1}{2r+1}\rfloor }^{\lfloor \frac{m-r}{2r+1}\rfloor }\left(\genfrac{}{}{0ex}{}{2m+1}{m-(2r+1)k-r}\right)& (n=2m+1)\end{array},$$

$${\delta }_{2r+1\mid n-2k}:=\{\begin{array}{ll}0& (2r+1\nmid n-2k)\\ 1& (2r+1\mid n-2k)\end{array}.$$

$$\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\left(\genfrac{}{}{0ex}{}{n-k+r-1}{k}\right)F_{n-2k+1}^{(r)}$$

$$=\frac{1}{2}\left((-1{)}^{\lfloor \frac{n-1-2k}{2r+1}\rfloor }-(-1{)}^{\lfloor \frac{n-2k-3}{2r+1}\rfloor }\right)=\{\begin{array}{ll}1& (n\equiv 0,1(\mathrm{mod}4r+2))\\ -1& (n\equiv 2r+1,2r+2(\mathrm{mod}4r+2))\\ 0& (\mathrm{others})\end{array},$$

$$2\sum _{k=0}^{\lfloor \frac{n+1}{2}\rfloor }\left(\genfrac{}{}{0ex}{}{2k-n-1}{k}\right)L_{n-2k}^{(r)}-\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\left(\genfrac{}{}{0ex}{}{2k-n}{k}\right)L_{n-2k}^{(r)}$$

$$=(-1{)}^n(-1+(2r+1){\delta }_{2r+1\mid n})=\{\begin{array}{ll}(-1{)}^{n-1}& (2r+1\nmid n)\\ (-1{)}^{n}2r& (2r+1\mid n)\end{array}.$$

$$F_1^{(r)}=1,F_0^{(r)}=F_{-1}^{(r)}=\cdots =F_{-r+2}^{(r)}=0,$$

$$L_n^{(r)}=\{\begin{array}{ll}-2^{2m-1}+\frac{2r+1}{2}\left(\genfrac{}{}{0ex}{}{2m}{m}\right)& (n=2m)\\ 4^m& (n=2m+1)\end{array}(n=0,1,\dots ,r-1).$$

$$a_{n+r}^{(r)}=\sum _{j=0}^{\lfloor \frac{r-1}{2}\rfloor }(-1{)}^j(\genfrac{}{}{0ex}{}{r-1-j}{j})a_{n+r-1-2j}^{(r)}+\sum _{j=0}^{\lfloor \frac{r-2}{2}\rfloor }(-1{)}^j\left(\genfrac{}{}{0ex}{}{r-1-j}{j+1}\right)a_{n+r-2-2j}^{(r)}.$$

$$\lambda \in {\mathcal{P}}_r:=\{\nu =({\nu }_1,\dots ,{\nu }_r)\in {\mathbb{Z}}_{\ge 0}^r\mid {\nu }_1\ge \dots \ge {\nu }_r\}$$

$$(n)$$

$$(1^n)$$

$$\sigma .\mathbf{z}:=(z_{\sigma (1)},\dots ,z_{\sigma (r)}).$$

$$s_{\lambda }(\mathbf{z})$$

Full text

1 Introduction

Throughout the paper, we denote the set of non-negative integers by $\mathbb{Z}_{\geq 0}$, the field of real numbers by $\mathbb{R}$, the field of complex numbers by $\mathbb{C}$ and $\mathbb{C}^{\times}:=\mathbb{C}\setminus\{0\}$. Let $z_{1},\ldots,z_{r}$ be $r$ independent variables and $\mathbf{z}:=(z_{1},\ldots,z_{r})$. For each non-negative integer $n$, the $n$th elementary symmetric polynomial $e_{n}^{(r)}$, complete homogeneous polynomials $h_{n}^{(r)}$ and power symmetric polynomial $p_{n}^{(r)}$ are defined by

[TABLE]

respectively. For $\mathbf{z}=(z_{1},\ldots,z_{r})\in{\mathbb{C}^{\times}}^{r}$, we put

[TABLE]

Our main results are new identities for these three types of symmetric polynomials $f=e,h,p$, which are relationships between $f_{n}^{(r)}(\mathbf{z}+\mathbf{z}^{-1})$ and $f_{n}^{(2r)}(\mathbf{z},\mathbf{z}^{-1})$. More precisely, we determine the following expansion coefficients $a_{n,k}^{(f)}$ and $b_{n,k}^{(f)}$,

[TABLE]

In this article we call (1.4) (resp. (1.5)) the first kind formulas (resp. the second kind formulas).

Theorem 1** (The first kind formulas).**

*For any non-negative integer $m$, we have the following identities.

(1)*

[TABLE]

where

[TABLE]

*and $\lfloor x\rfloor$ is the greatest integer not exceeding $x\in\mathbb{R}$.

(2)*

[TABLE]

(3)**

[TABLE]

Theorem 2** (The second kind formulas).**

(1)* For $n=0,1,\ldots,2r$, we have*

[TABLE]

(2)* For any non-negative integer $n$,*

[TABLE]

(3)* For any positive integer $n$,*

[TABLE]

The proofs of (1.6) and (1.7) are more difficult than (1.9) and (1.10). In particular we need a hypergeometric identity (2.20) to derive the explicit formulas of $a_{n,k}^{(e)}$ or $a_{n,k}^{(h)}$. The second kind formulas and (1.8) which are proved immediately by the binomial formula and generating functions may be well known. However we have not found appropriate references about these formulas and their interesting applications which are to propose some new formulas of Fibonacci and Lucas numbers etc. In particular, we apply Theorem 1 and Theorem 2 to some special values of $h_{n}^{(r)}$ and $p_{n}^{(r)}$;

[TABLE]

where

[TABLE]

Since the case of $r=1$ is trivial sequences

[TABLE]

and the case of $r=2$ is the classical Fibonacci numbers $\{F_{n+1}\}_{n}$ and Lucas numbers $\{L_{n}\}_{n}$

[TABLE]

the sequences $\{F_{n+1}^{(r)}\}_{n}$ and $\{L_{n}^{(r)}\}_{n}$ are regarded as a higher order analogue of the classical Fibonacci and Lucas numbers. From Theorem 1, Theorem 2 and some fundamental results for symmetric functions, we propose the following fundamental formulas for $F_{n+1}^{(r)}$ and $L_{n}^{(r)}$.

Theorem 3** (Explicit formulas of $F_{n}^{(r)}$ and $L_{n}^{(r)}$).**

[TABLE]

where

[TABLE]

Theorem 4** (Inversion of the explicit formulas).**

[TABLE]

Theorem 5** (Initial values and recurrence formulas).**

Initial values of $\{F_{n+1}^{(r)}\}_{n\in\mathbb{Z}}$ and $\{L_{n}^{(r)}\}_{n\in\mathbb{Z}}$ are given by

[TABLE]

The sequences $\{F_{n+1}^{(r)}\}_{n\in\mathbb{Z}}$ and $\{L_{n}^{(r)}\}_{n\in\mathbb{Z}}$ satisfy the following same recursion:

[TABLE]

The content of this article is as follows. In Section 2, we refer some basic formulas for symmetric polynomials and the Gauss hypergeometric function from [AAR] and [M]. Section 3 is the main part of this article. In this section, we prove Theorem 1 and Theorem 2, and give their principal specialization. In Section 4, we evaluate $e_{n}^{(2r)}\left(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota}\right)$, $h_{n}^{(2r)}\left(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota}\right)$ and $p_{n}^{(2r)}\left(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota}\right)$ by the definitions and generating functions. By substituting these evaluations into Theorem 1 and Theorem 2, we propose Theorem 3, Theorem 4 and Theorem 5. Further, we consider some specializations of our main results and drive some interesting binomial sum formulas, including new formulas for the Fibonacci and Lucas numbers (see Corollary 18 and Corollary 19). In Appendix, we mention some congruence properties and other formulas (generating functions, determinant formulas, some relations) for $F_{n+1}^{(r)}$ and $L_{n}^{(r)}$ that are proven independently of Theorem 1 and Theorem 2.

2 Preliminaries

2.1 Symmetric polynomials

Refer to Macdonald [M] for the details in this subsection. We fix a positive integer $r$, and denote the partition set of length $r$ by

[TABLE]

and the symmetric group of degree $r$ by $\mathfrak{S}_{r}$. For some special partitions, we use the following notations

[TABLE]

The symmetric group $\mathfrak{S}_{r}$ acts on $\mathbf{z}=(z_{1},\ldots,z_{r})\in\mathbb{C}^{r}$ by

[TABLE]

For any partition $\lambda$, we define Schur polynomial $s_{\lambda}(\mathbf{z})$ and monomial symmetry polynomial $m_{\lambda}(\mathbf{z})$ by

[TABLE]

where $\mathop{\mathrm{det}}\,$ is the usual determinant and

[TABLE]

We remark that Schur polynomial extends to Schur function $s_{\lambda}(\mathbf{z})$ for $\lambda=(\lambda_{1},\ldots,\lambda_{r})\in\mathbb{C}^{r}$ by (2.1).

It is well known that

[TABLE]

which is Schur $s_{(1^{n})}$ or monomial $m_{(1^{n})}$ with one column are $e_{n}^{(r)}$, and monomial $m_{(n)}$ with one row is $p_{n}^{(r)}$, and Schur $s_{(n)}$ with one row is $h_{n}^{(r)}$ ([M] Chapter I Section 3 (3.9)). From (2.5), we extend the complete homogeneous polynomials to $h_{n}^{(r)}(\mathbf{z})$ $(n\in\mathbb{Z})$: namely

[TABLE]

By this extension (2.6) and the definition of Schur function, for any $r$ we have

[TABLE]

We list up some required formulas for symmetric polynomials in [M].

Lemma 6**.**

(1)* Generating functions*

[TABLE]

(2)* $q$-binomial formula*

[TABLE]

where $\binom{n}{k}_{q}$ denotes the $q$-binomial coefficient

[TABLE]

(3)* Wronski relation and Newton’s formula *

[TABLE]

Actually, (2.8) is [M] p19 (2.2) and (2.9) is [M] p21 (2.5) exactly. For (2.11) and (2.12), see [M] p26 Examples 3. Similarly, (2.13) and (2.14) are [M] p21 (2.6*′) and p23 (2.11′*) respectively.

2.2 The Gauss hypergeometric function

Let $a,b,c,z$ be complex numbers such that $c$ is not non-negative integers, and $(a)_{m}$ be the raising factorial defined by

[TABLE]

We recall Gauss hypergeometric function

[TABLE]

and for any complex numbers $\alpha$ and $x$ we put

[TABLE]

Since

[TABLE]

the function $\psi(\alpha;x)$ is analytically continued to $4x\in\mathbb{C}\setminus\{0,1\}$ by analytic continuation of ${{}_{2}F_{1}}(z)$.

Lemma 7**.**

(1)* Another expression*

[TABLE]

(2)* Closed form*

[TABLE]

(3)* Index law*

[TABLE]

(4)* Quadratic formula*

[TABLE]

Proof.

(1) By the definition of $\psi(\alpha;x)$ and $c_{a,k}$, we have

[TABLE]

(2) We remark a hypergeometric transformation [AAR] (3.1.10)

[TABLE]

Thus,

[TABLE]

The formulas (2.18) and (2.19) follow from (2.17) immediately. ∎

The following Lemma is a corollary of Lemma 7 and the key step in the proof of Theorem 1.

Lemma 8**.**

If

[TABLE]

then

[TABLE]

and for any non-negative integer $N$

[TABLE]

Proof.

By quadratic formula,

[TABLE]

Thus, we have

[TABLE]

Here the second and third equalities follow from (2.18) and (2.16) respectively. ∎

Remark 9**.**

We mention some properties of $c_{n,k}$. For non-negative integers $n$ and $k$, we make the table of $c_{n,k}>0$.

This table is determined exactly by initial conditions

[TABLE]

and a recursion formula

[TABLE]

The sequence $c_{n,k}$ is a kind of Clebsch-Gordan coefficients for the Lie algebra $sl_{2}$. In fact, from the above initial conditions and recursion of $c_{n,k}$, we have

[TABLE]

that is the classical Clebsch-Gordan rule for $sl_{2}$ exactly.

Further $c_{n,k}$ is also a typical example of Kostka numbers $K_{\lambda\mu}$ (see [S] Chapter 2 Section 2.11 and Chapter 4 Section 4.9). We remark Young’s rule

[TABLE]

and

[TABLE]

By putting $r=2$, $\mu_{1}=\cdots=\mu_{n}=1,z_{1}=e^{\sqrt{-1}\theta},z_{2}=e^{-\sqrt{-1}\theta}$ in (2.22), we have

[TABLE]

Finally, by comparing (2.21) and (2.23), we have

[TABLE]

3 Proofs of Theorem 1 and Theorem 2

From (2.8), (2.9) and simple calculation

[TABLE]

we obtain the following key lemma.

Lemma 10**.**

If

[TABLE]

*then *

[TABLE]

By Lemma 10 and the definition of the power symmetric polynomials, we prove Theorem 1 and Theorem 2.

Proof of Theorem 1 (1) By (3.1),

[TABLE]

where

[TABLE]

Hence, from (2.20) we have

[TABLE]

By comparing coefficients of (3.3) and (3.4), we obtain the conclusion.

(2) Similarly, from (3.2)

[TABLE]

and (2.20) we have

[TABLE]

The formula (1.7) follows from (3.5) and (3.6).

(3) We prove this formula without generating function and other Lemmas. In fact, by applying the usual binomial formula we have

[TABLE]

∎

Proof of Theorem 2 (1) From (3.1) and binomial theorem, we have

[TABLE]

(2) Similarly, we have

[TABLE]

(3) It is enough to show that the case of $r=1$ which is

[TABLE]

From (3.1) and simple calculus,

[TABLE]

If $|zy|<1$, $|z^{-1}y|<1$ and $|y|<1$, then

[TABLE]

∎

Finally we consider the principal specialization of Theorem 1 and Theorem 2, which means substituting

[TABLE]

for $\mathbf{z}$. In this special case, we evaluate $e_{n}^{(2r)}(\boldsymbol{q}^{+\iota},\boldsymbol{q}^{-\iota})$, $h_{n}^{(2r)}(\boldsymbol{q}^{+\iota},\boldsymbol{q}^{-\iota})$ and $p_{n}^{(2r)}(\boldsymbol{q}^{+\iota},\boldsymbol{q}^{-\iota})$ explicitly.

Proposition 11**.**

*For any non-negative integer $n$, we have the following identities.

(1)*

[TABLE]

(2)**

[TABLE]

(3)**

[TABLE]

Proof.

(1) From generating function of elementary symmetric polynomials (2.8),

[TABLE]

By $q$-binomial formula (2.11), we have

[TABLE]

(2) Similarly, we have

[TABLE]

(3) By the definition of power sum and geometric series, we have

[TABLE]

We remark this formula holds on the limit $q\rightarrow 1$. ∎

Corollary 12**.**

(1)* For any non-negative integer $m$,*

[TABLE]

For $n=0,1,\ldots,2r$,

[TABLE]

(2)* For any non-negative integers $n$ and $m$, we have*

[TABLE]

(3)* For any non-negative integer $m$,*

[TABLE]

For any positive integer $n$,

[TABLE]

4 Applications to $F_{n}^{(r)}$ and $L_{n}^{(r)}$

In this section we investigate more specializations of Theorem 1 and Theorem 2, and prove Theorem 3, Theorem 4 and Theorem 5. To apply Theorem 1 and Theorem 2 to $F_{n}^{(r)}$ and $L_{n}^{(r)}$, we evaluate $e_{n}^{(2r)}(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota})$, $h_{n}^{(2r)}(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota})$, $p_{n}^{(2r)}(-\boldsymbol{\zeta}^{+\iota},-\boldsymbol{\zeta}^{-\iota})$ and $e_{m}^{(r)}(-\boldsymbol{\zeta}^{+\iota}-\boldsymbol{\zeta}^{-\iota})$.

Proposition 13**.**

(1)* For $n=0,1,\ldots,2r$, we have*

[TABLE]

(2)* For any non-negative integer $n$,*

[TABLE]

(3)* For any non-negative integer $n$,*

[TABLE]

where

[TABLE]

Proof.

(1) From (2.8), we have

[TABLE]

(2) From (2.9), we have

[TABLE]

(3) By the definition of $p_{n}^{(2r)}$,

[TABLE]

∎

Proof of Theorem 3 and Theorem 4 From Theorem 1 (2) and (3) and Proposition 1 (2) and (3), we derive explicit formulas of $F_{n}^{(r)}$ and $L_{n}^{(r)}$. Similarly, Theorem 4 follows from Theorem 1 (2) and (3) and Proposition 1 (2) and (3). ∎

By specialization of Theorem 3 (2) and (3), we obtain the initial values of $F_{n+1}^{(r)}$ and $L_{n}^{(r)}$.

Corollary 14**.**

(1)* If $m\leq r$, then we have*

[TABLE]

(2)* If $m<2r+1$, then we have*

[TABLE]

If $m<r$, then we have

[TABLE]

To prove Theorem 5, we need to evaluate $e_{m}^{(r)}(-\boldsymbol{\zeta}^{+\iota}-\boldsymbol{\zeta}^{-\iota})$, which can be computed from (1.6) and (4.1).

Proposition 15**.**

For $m=0,1,\ldots,r$, we have

[TABLE]

Remark 16**.**

(1) For (4.7), we give another proof without using (1.6). Let $x:=z+z^{-1}$. First, we remark

[TABLE]

On the other hand, if $|u|<|z|<|u|^{-1}$, then

[TABLE]

Hence we obtain the conclusion (4.7).

(2) The formula (4.7) is obtained by substituting (4.1) into (1.6). Similarly, by substituting (4.1) for (1.9), for $n=0,1,\ldots,2r$ we obtain

[TABLE]

Proof of Theorem 5 From the Wronski relations, Newton’s formulas and (4.7), for any non-negative integer $n$ we have

[TABLE]

Then $\{F_{n+1}^{(r)}\}_{n\geq 0}$ and $\{L_{n}^{(r)}\}_{n\geq 0}$ satisfy the recursion (1.20).

The initial values of $\{F_{n+1}^{(r)}\}_{n\geq 0}$ are determined by the vanishing property (2.7)

[TABLE]

and $F_{1}^{(r)}=h_{0}^{(r)}=1$. The initial values of $\{L_{n}^{(r)}\}_{n\geq 0}$ follows from Corollary 14 (2).∎

Example 17**.**

$r=1$

[TABLE]

$r=2$ (Fibonacci numbers and Lucas numbers)

[TABLE]

$r=3$ (OEIS A006053 and OEIS A096975)

[TABLE]

$r=4$ (OEIS A188021 and OEIS A094649)

[TABLE]

$r=5$ (OEIS A231181 and OEIS A189234)

[TABLE]

$r=6$

[TABLE]

Finally, we mention some interesting examples of our results, including the seemingly new formulas for the Fibonacci and Lucas numbers.

Corollary 18**.**

For any non-negative integers $m$ and $n$, we have

[TABLE]

and

[TABLE]

The formula (LABEL:eq:Andrews_Fibonacci_formula) was given by Andrews [A].

Corollary 19**.**

For any non-negative integer $m$, we have

[TABLE]

For any positive integer $n$, we have

[TABLE]

Appendix A Some congruence relations for $F_{n}^{(r)}$ and $L_{n}^{(r)}$

All of the results so far have been obtained as specializations of Theorem 1 and Theorem 2, but in this section, we mention some properties for $F_{n}^{(r)}$ and $L_{n}^{(r)}$ that can be obtained independently of Theorem 1 and Theorem 2.

Theorem 20**.**

Let $p:=2r+1$ be a prime number. If $q$ is a odd prime number such that $q\equiv\pm 1\,\,(\mathrm{mod}\,p)$, then

[TABLE]

In particular, for any non-negative integer $k$ we have

[TABLE]

Proof.

Let $\mathbb{F}_{q}$ be a finite field order $q$ and $\zeta_{p}$ be a primitive $p$-th root of unity. Since the both side of (A.1) and (A.2) are integers, it is enough to show that the equalities in $\mathbb{F}_{q}[\zeta_{p}]$. For any integer $i$ a simple calculation shows that

[TABLE]

Hence by the definition of $L_{n}^{r}$ we obtain

[TABLE]

To prove (A.1), we need the discriminant of $-\zeta_{p}-\zeta_{p}^{-1}$ [L] Theorem 3.8

[TABLE]

From this evaluation, we have

[TABLE]

We point out even if $r$ is even then $p^{\frac{r-1}{2}}\in\mathbb{F}_{p}$ by the first supplement to quadratic reciprocity. Thus we have

[TABLE]

Here the first equality follows from (2.5). Finally, since $p$ does not divide $q$, we obtain (A.1).

The formulas (A.3) - (A.7) follow from (A.1), (A.2) and Corollary 14 immediately. ∎

Appendix B Other formulas for $F_{n}^{(r)}$ and $L_{n}^{(r)}$ from symmetric polynomials

Since the sequences $\{F_{n}^{(r)}\}_{n}$ and $\{L_{n}^{(r)}\}_{n}$ are special values of $h_{n}^{(r)}(\mathbf{z})$ and $p_{n}^{(r)}(\mathbf{z})$ respectively, various formulas for $F_{n}^{(r)}$ and $L_{n}^{(r)}$ are derived immediately from specializations of some formulas for symmetric polynomials [M]. In this section, we list some typical formulas obtained from symmetric polynomials.

Generating functions

[TABLE]

Generating functions (B.1) and (B.2) are obtained by substituting (4.7) into (2.5) and (2.10) in [M].

Determinant formulas For convenience, put

[TABLE]

and

[TABLE]

From (2.5), (A.8) and the determinant formulas on p28 of [M], we obtain the following determinant formulas for $F_{n}^{(r)}$ and $L_{n}^{(r)}$.

[TABLE]

Some relations For any partition $\lambda$ let $z_{\lambda}$ denote the product

[TABLE]

where $m_{i}=m_{i}(\lambda)$ is the number of parts of $\lambda$ equal to $i$. Then we have

[TABLE]

where $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ run over partitions and $|\lambda|$ denote the sum of the parts

[TABLE]

These formulas follow from (2.11) and (2.14*′*) in [M].