# New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The New Generalized Polynomials: Definitions and Properties

**Remark**

**1.**

## 3. Partial Differential Equations for Polynomials in (13)

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

- (i)
- For $m+n-1\le j\le m+n+c-1,$ then we obtain$$(j+1){S}_{j+1}={x}^{k}\sum _{l=0}^{j}{S}_{j-l}{S}_{l}+{y}^{m}(n+m)\sum _{l=0}^{j-m-n+1}{S}_{l}{S}_{j-m-n-l+1}.$$
- (ii)
- For $j\le m+n-1,$ then we derive$$(j+1){S}_{j+1}={x}^{k}\sum _{l=0}^{j}{S}_{j-l}{S}_{l}.$$
- (iii)
- For $j\ge m+n+c-1,$ then we get$$\begin{array}{cc}\hfill (j+1){S}_{j+1}=& \phantom{\rule{4pt}{0ex}}{x}^{k}\sum _{l=0}^{j}{S}_{j-l}{S}_{l}+{y}^{m}(n+m)\sum _{l=0}^{j-m-n+1}{S}_{l}{S}_{j-m-n-l+1}\hfill \\ & \phantom{\rule{4pt}{0ex}}+{z}^{c}(n+m+c)\sum _{l=0}^{j-m-n-c+1}{S}_{l}{S}_{j-m-n-c-l+1}.\hfill \end{array}$$

**Theorem**

**5.**

## 4. Some Applications of Generating Functions

**Case****1.**- Taking $t=\frac{1}{a}$ in (15) for $\left|a\right|>1$, we get the following equation$$\sum _{j=0}^{\infty}\frac{{W}_{j}}{{a}^{j}}=\frac{{a}^{m+c}}{{a}^{m+n+c}-{x}^{k}{a}^{m+n+c-1}-{y}^{m}{a}^{c}-{z}^{c}}.$$

- (
**i**) - Substituting $a=2,$ $x\to {x}^{2}$, $y\to x,$ $z\to 1$ and $k=m=n=c=1$ in (22), we obtain the relation for the tribonacci polynomials as$$\sum _{j=0}^{\infty}\frac{{t}_{j}(x)}{{2}^{j}}=\frac{4}{7-4{x}^{2}-2x}.$$Writing $x=1$ in (23), we have$$\sum _{j=0}^{\infty}\frac{{T}_{j}}{{2}^{j}}=4\phantom{\rule{0.166667em}{0ex}},$$
- (
**ii**) - (
**iii**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=2$, and $k=m=n=c=1$ into (22), we get for the Fibonacci polynomials$$\sum _{j=0}^{\infty}\frac{{F}_{j}(x)}{{2}^{j}}=\frac{2}{3-2x},$$$$\sum _{j=0}^{\infty}\frac{{F}_{j}}{{2}^{j}}=2\phantom{\rule{0.166667em}{0ex}},$$
- (
**iv**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=3$, and $k=m=n=c=1$ in (22), we get for the Fibonacci polynomials$$\sum _{j=0}^{\infty}\frac{{F}_{j}(x)}{{3}^{j+1}}=\frac{1}{8-3x}.$$
- (
**v**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=8$, and $k=m=n=c=1$ in (22), we get for the Fibonacci polynomials$$\sum _{j=0}^{\infty}\frac{{F}_{j}(x)}{{8}^{j+1}}=\frac{1}{63-3x}.$$
- (
**vi**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=-10$, and $k=m=n=c=1$ in (22), we get for the Fibonacci polynomials$$\sum _{j=0}^{\infty}\frac{{F}_{j}(x)}{{(-10)}^{j+1}}=\frac{1}{99+10x}.$$
- (
**vii**) - Substituting $x\to 2x,$ $y\to 1,$ $z\to 0$, $a=3,$ and $k=m=n=c=1$ in (22), we get$$\sum _{j=0}^{\infty}\frac{{P}_{j}(x)}{{3}^{j+1}}=\frac{1}{8-6x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{P}_{j}}{{3}^{j+1}}=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}},$$
- (
**viii**) - Substituting $x\to 1,$ $y\to 2y,$ $z\to 0,$ $a=3,$ and $k=m=n=c=1$ in (22), we get$$\sum _{s=0}^{\infty}\frac{{J}_{s}(x)}{{3}^{s+1}}=\frac{1}{6-2y}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{s=0}^{\infty}\frac{{J}_{s}}{{3}^{s+1}}=\frac{1}{4}\phantom{\rule{0.166667em}{0ex}},$$

**Case****2.**- Taking $t=\frac{1}{a}$ in (16) for $\left|a\right|>1$, we get the following equation$$\sum _{j=0}^{\infty}\frac{{K}_{j}}{{a}^{j}}=\frac{{a}^{m+n+c}\alpha (t;x,y)-{a}^{m+c}\beta (t;x,y)}{{a}^{m+n+c}-{x}^{k}{a}^{m+n+c-1}-{y}^{m}{a}^{c}-{z}^{c}}.$$

- (
**i**) - Substituting $x\to {x}^{2},$ $y\to x,$ $z\to 1$, $a=2$, and $k=m=n=c=1,$ $\alpha (t;x,y)=3,$ $\beta (t;x,y)=2{x}^{2}+xt$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{k}_{j}(x)}{{2}^{j}}=\frac{24-8{x}^{4}-2{x}^{2}}{7-4{x}^{2}-2x},$$$$\sum _{j=0}^{\infty}\frac{{k}_{j}}{{2}^{j+1}}=7,$$
- (
**ii**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=2$, and $k=m=n=c=1,$ $\alpha (t;x,y)=2,$ $\beta (t;x,y)=x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{L}_{j}(x)}{{2}^{j+1}}=\frac{4-x}{3-2x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{L}_{j}}{{2}^{j+1}}=3,$$
- (
**iii**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=10$, and $k=m=n=c=1,$ $\alpha (t;x,y)=2,$ $\beta (t;x,y)=x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{L}_{j}(x)}{{10}^{j}}=\frac{200-10x}{99-10x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{L}_{j}}{{10}^{j+1}}=\frac{19}{89}=\frac{{L}_{6}-{L}_{1}}{{F}_{11}}$$
- (
**iv**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=3$, and $k=m=n=c=1,$ $\alpha (t;x,y)=2,$ $\beta (t;x,y)=x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{L}_{j}(x)}{{3}^{j+1}}=\frac{6-x}{8-3x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{L}_{j}}{{3}^{j+1}}=1.$$
- (
**v**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=8$, and $k=m=n=c=1$, $\alpha (t;x,y)=2,$ $\beta (t;x,y)=x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{L}_{j}(x)}{{8}^{j+1}}=\frac{16-x}{63-8x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{L}_{j}}{{8}^{j+1}}=\frac{3}{11}=\frac{{L}_{2}}{{L}_{5}}\phantom{\rule{0.166667em}{0ex}}.$$
- (
**vi**) - Substituting $x\to x,$ $y\to 1,$ $z\to 0$, $a=-10$, and $k=m=n=c=1,$ $\alpha (t;x,y)=2,$ $\beta (t;x,y)=x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{L}_{j}(x)}{{(-10)}^{j+1}}=\frac{-20-x}{99+10x}\phantom{\rule{0.166667em}{0ex}}.$$
- (
**vii**) - Substituting $x\to 2x,$ $y\to 1,$ $z\to 0$, $a=5$, and $k=m=n=c=1,$ $\alpha (t;x,y)=2,$ $\beta (t;x,y)=2x$ in (31), we get$$\sum _{j=0}^{\infty}\frac{{Q}_{j}(x)}{{5}^{j+1}}=\frac{5-x}{12-5x}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{j=0}^{\infty}\frac{{Q}_{j}}{{5}^{j+1}}=\frac{4}{7}\phantom{\rule{0.166667em}{0ex}},$$
- (
**viii**) - Substituting $x\to 1,$ $y\to 2y,$ $z\to 0$, $a=3$, and $k=m=n=c=1$, $\alpha (t;x,y)=2,$ $\beta (t;x,y)=1$ in (31), we get$$\sum _{s=0}^{\infty}\frac{{j}_{s}(y)}{{3}^{s+1}}=\frac{5}{6-2y}\phantom{\rule{0.166667em}{0ex}},$$$$\sum _{s=0}^{\infty}\frac{{j}_{s}}{{3}^{s+1}}=\frac{5}{4}\phantom{\rule{0.166667em}{0ex}},$$
- (
**ix**)

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Koshy, T. Fibonacci and Lucas Numbers with Applications; John Wiley and Sons Inc.: New York, NY, USA, 2001. [Google Scholar]
- Nalli, A.; Haukkanen, P. On generalized Fibonacci and Lucas polynomials. Chaos Solitons Fractals
**2009**, 42, 3179–3186. [Google Scholar] [CrossRef] - Vajda, S. Fibonacci and Lucas Numbers, and the Golden Section, Theory and Applications; Ellis Horwood Limited: Chichester, UK, 1989. [Google Scholar]
- Elia, M. Derived sequences, the tribonacci recurrence and cubic forms. Fibonacci Q.
**2001**, 39, 107–109. [Google Scholar] - Feinberg, M. Fibonacci-tribonacci. Fibonacci Q.
**1963**, 3, 70–74. [Google Scholar] - Hoggatt, V.E., Jr.; Bicknell, M. Generalized Fibonacci polynomials. Fibonacci Q.
**1973**, 11, 457–465. [Google Scholar] - Tan, M.; Zhang, Y. A note on bivariate and trivariate Fibonacci polynomials. Southeast Asian Bull. Math.
**2005**, 29, 975–990. [Google Scholar] - Kocer, E.G.; Gedikli, H. Trivariate Fibonacci and Lucas polynomials. Konuralp J. Math.
**2016**, 4, 247–254. [Google Scholar] - Tuglu, N.; Kocer, E.G.; Stakhov, A. Bivariate Fibonacci like p-polynomials. Appl. Math. Comput.
**2011**, 217, 10239–10246. [Google Scholar] [CrossRef] - Kim, T.; Kim, D.S.; Dolgy, D.V.; Park, J.W. Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials. J. Inequal. Appl.
**2018**, 148. [Google Scholar] [CrossRef] [PubMed] - Kim, T.; Dolgy, D.V.; Kim, D.S.; Seo, J.J. Convolved Fibonacci numbers and their applications. arXiv, 2017; arXiv:1607.06380. [Google Scholar]
- Wang, T.; Zhang, W. Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roum.
**2012**, 55, 95–103. [Google Scholar] - Wu, Z.; Zhang, W. Several identities involving the Fibonacci polynomials and Lucas polynomials. J. Inequal. Appl.
**2013**, 2013, 14. [Google Scholar] [CrossRef] - Ozdemir, G.; Simsek, Y. Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers. Filomat
**2016**, 30, 969–975. [Google Scholar] [CrossRef] - Lee, G.Y.; Asci, M. Some properties of the (p,q)-Fibonacci and (p,q)-Lucas polynomials. J. Appl. Math.
**2012**, 2012, 264842. [Google Scholar] [CrossRef] - Horadam, A.F. Chebyshev and Fermat polynomials for diagonal functions. Fibonacci Q.
**1979**, 17, 328–333. [Google Scholar] - Lidl, R.; Mullen, G.; Tumwald, G. Dickson Polynomials. In Pitman Monographs and Surveys in Pure and Applied Mathematics; Longman Scientific and Technical: Essex, UK, 1993; Volume 65. [Google Scholar]
- Shannon, A.G.; Horadam, A.F. Some relationships among Vieta, Morgan-Voyce and Jacobsthal polynomials. In Applications of Fibonacci Numbers; Howard, F.T., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999; pp. 307–323. [Google Scholar]
- Cheon, G.-S.; Kim, H.; Shapiro, L.W. A generalization of Lucas polynomial sequence. Discret. Appl. Math.
**2009**, 157, 920–927. [Google Scholar] [CrossRef] - Bulut, H.; Pandir, Y.; Baskonus, H.M. Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order. AIP Conf. Proc.
**2013**, 1558, 1914–1918. [Google Scholar] - Mirzaee, F.; Hoseini, S.F. Solving singularly perturbed differential-difference equations arising in science and engineering whit Fibonacci polynomials. Results Phys.
**2013**, 3, 134–141. [Google Scholar] [CrossRef] - Mirzaee, F.; Hoseini, S.F. Solving systems of linear Fredholm integro-differential equations with Fibonacci polynomials. Ain. Shams Eng.
**2014**, 5, 271–283. [Google Scholar] [CrossRef] [Green Version] - Kurt, A.; Yalinbash, S.; Sezer, M. Fibonacci-collocation method for solving high-order linear Fredholm integro-differential-difference equations. Int. J. Math. Math. Sci.
**2013**, 2013, 486013. [Google Scholar] [CrossRef]

x | y | z | k | m | n | c | Special Case |
---|---|---|---|---|---|---|---|

x | y | z | 1 | 1 | 1 | 1 | Trivariate Fibonacci Polynomials [8] |

${x}^{2}$ | x | 1 | 1 | 1 | 1 | 1 | tribonacci Polynomials [8] |

x | y | 0 | 1 | 1 | 1 | c | Bivariate Fibonacci Polynomials [9] |

x | 1 | 0 | 1 | p | 1 | c | Fibonacci $p-$Polynomials [9] |

$2x$ | 1 | 0 | 1 | p | 1 | c | Pell $p-$Polynomials [9] |

x | 1 | 0 | 1 | 1 | 1 | c | Fibonacci Polynomials [9] |

$2x$ | 1 | 0 | 1 | 1 | 1 | c | Pell Polynomials [9] |

1 | $2y$ | 0 | k | 1 | 1 | c | Jacobsthal Polynomials [9] |

$3x$ | $-2$ | 0 | 1 | 1 | 1 | c | Fermat Polynomials [15] |

x | $-2$ | 0 | 1 | 1 | 1 | c | First kind of Fermat–Horadam Polynomials [16] |

x | $-\alpha $ | 0 | 1 | 1 | 1 | c | Second kind of Dickson Polynomials [17] |

$x+2$ | $-1$ | 0 | 1 | 1 | 1 | c | Morgan–Voyce Polynomials [18] |

$x+1$ | $-x$ | 0 | 1 | 1 | 1 | c | Delannoy Polynomials [19] |

$h(x)$ | 1 | 0 | 1 | 1 | 1 | c | $h(x)-$Fibonacci Polynomials [2] |

$p(x)$ | $q(x)$ | 0 | 1 | 1 | 1 | c | $(p,q)-$Fibonacci Polynomials [15] |

1 | 1 | 0 | k | 1 | 1 | c | Fibonacci Numbers [9] |

2 | 1 | 0 | 1 | 1 | 1 | c | Pell Numbers [9] |

1 | 2 | 0 | k | 1 | 1 | c | Jacobsthal Numbers [9] |

$\mathit{\alpha}$ | $\mathit{\beta}$ | x | y | z | k | m | n | c | Special Case |
---|---|---|---|---|---|---|---|---|---|

3 | $2x+yt$ | x | y | z | 1 | 1 | 1 | 1 | Trivariate Lucas Polynomials [8] |

3 | $2{x}^{2}+xt$ | ${x}^{2}$ | x | 1 | 1 | 1 | 1 | 1 | tribonacci-Lucas Polynomials [8] |

2 | $xz$ | x | y | 0 | 1 | 1 | 1 | c | Bivariate Lucas Polynomials [9] |

$p+1$ | $px$ | x | 1 | 0 | 1 | p | 1 | c | Lucas $p-$Polynomials [9] |

0 | $-1$ | $2x$ | 1 | 0 | 1 | p | 1 | c | Pell Lucas $p-$Polynomials [9] |

2 | x | x | 1 | 0 | 1 | 1 | 1 | c | Lucas Polynomials [9] |

2 | $2x$ | $2x$ | 1 | 0 | 1 | 1 | 1 | c | Pell Lucas Polynomials [9] |

2 | 1 | 1 | $2y$ | 0 | k | 1 | 1 | c | Jacobsthal Lucas Polynomials [9] |

2 | $3x$ | $3x$ | $-2$ | 0 | 1 | 1 | 1 | c | Fermat Lucas Polynomials [15] |

2 | x | x | $-2$ | 0 | 1 | 1 | 1 | c | Second kind of Fermat–Horadam P. [16] |

2 | x | x | $-\alpha $ | 0 | 1 | 1 | 1 | c | First kind of Dickson Polynomials [17] |

2 | $x+2$ | $x+2$ | $-1$ | 0 | 1 | 1 | 1 | c | Morgan–Voyce Polynomials [18] |

2 | $x+1$ | $x+1$ | $-x$ | 0 | 1 | 1 | 1 | c | Corona Polynomials [19] |

2 | $h(x)$ | $h(x)$ | 1 | 0 | 1 | 1 | 1 | c | $h(x)-$Lucas Polynomials [2] |

2 | $p(x)$ | $p(x)$ | $q(x)$ | 0 | 1 | 1 | 1 | c | $(p,q)-$Lucas Polynomials [15] |

2 | 1 | 1 | 1 | 0 | k | 1 | 1 | c | Lucas Numbers [9] |

2 | 2 | 2 | 1 | 0 | 1 | 1 | 1 | c | Pell–Lucas Numbers [9] |

2 | 1 | 1 | 2 | 0 | k | 1 | 1 | c | Jacobsthal–Lucas Numbers [9] |

t | t | 2 | 2 | $-1$ | 1 | 1 | 1 | 1 | Squares of Fibonacci Numbers [1] |

a | x | y | z | Formulas |
---|---|---|---|---|

2 | ${x}^{2}$ | x | 1 | ${\displaystyle \sum _{j=0}^{\infty}}}\frac{{t}_{j}(x)}{{2}^{j}}=\frac{4}{7-4{x}^{2}-2x$ |

2 | 1 | 1 | 1 | $\sum _{j=0}^{\infty}}\frac{{T}_{j}}{{2}^{j}}=4$ |

10 | ${x}^{2}$ | x | 1 | $\sum _{j=0}^{\infty}}\frac{{T}_{j}(x)}{{10}^{j+2}}=\frac{1}{999-100{x}^{2}-10x$ |

10 | 1 | 1 | 1 | $\sum _{j=0}^{\infty}}\frac{{T}_{j}}{{10}^{j+2}}=\frac{1}{889$ |

2 | x | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}(x)}{{2}^{j}}=\frac{2}{3-2x$ |

2 | 1 | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}}{{2}^{j}}=2$ |

3 | x | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}(x)}{{3}^{j+1}}=\frac{1}{8-3x$ |

3 | 1 | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}}{{3}^{j+1}}=\frac{1}{5}=\frac{1}{{F}_{5}$ |

8 | x | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}(x)}{{8}^{j+1}}=\frac{1}{63-3x$ |

8 | 1 | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}}{{8}^{j+1}}=\frac{1}{55}=\frac{1}{{F}_{10}$ |

$-10$ | x | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}(x)}{{(-10)}^{j+1}}=\frac{1}{99+10x$ |

$-10$ | 1 | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{F}_{j}}{{(-10)}^{j+1}}=\frac{1}{109$ |

3 | $2x$ | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{P}_{j}(x)}{{3}^{j+1}}=\frac{1}{8-6x$ |

3 | 2 | 1 | 0 | $\sum _{j=0}^{\infty}}\frac{{P}_{j}}{{3}^{j+1}}=\frac{1}{2$ |

3 | 1 | $2y$ | 0 | $\sum _{s=0}^{\infty}}\frac{{J}_{s}(x)}{{3}^{s+1}}=\frac{1}{6-2y$ |

3 | 1 | 2 | 0 | $\sum _{s=0}^{\infty}}\frac{{J}_{s}}{{3}^{s+1}}=\frac{1}{4$ |

a | x | y | z | $\mathit{\alpha}$ | $\mathit{\beta}$ | Formulas |
---|---|---|---|---|---|---|

2 | ${x}^{2}$ | x | 1 | 3 | $2{x}^{2}+xt$ | $\sum _{j=0}^{\infty}}\frac{{k}_{j}(x)}{{2}^{j}}=\frac{24-8{x}^{4}-2{x}^{2}}{7-4{x}^{2}-2x$ |

2 | 1 | 1 | 1 | 3 | $2+t$ | $\sum _{j=0}^{\infty}}\frac{{k}_{j}}{{2}^{j+1}}=7$ |

2 | x | 1 | 0 | 2 | x | $\sum _{j=0}^{\infty}}\frac{{L}_{j}(x)}{{2}^{j+1}}=\frac{4-x}{3-2x$ |

2 | 1 | 1 | 0 | 2 | 1 | $\sum _{j=0}^{\infty}}\frac{{L}_{j}}{{2}^{j+1}}=3$ |

10 | x | 1 | 0 | 2 | x | $\sum _{j=0}^{\infty}}\frac{{L}_{j}(x)}{{10}^{j}}=\frac{200-10x}{99-10x$ |

10 | 1 | 1 | 0 | 2 | 1 | $\sum _{j=0}^{\infty}}\frac{{L}_{j}}{{10}^{j+1}}=\frac{19}{89}=\frac{{L}_{6}-{L}_{1}}{{F}_{11}$ |

3 | x | 1 | 0 | 2 | x | $\sum _{j=0}^{\infty}}\frac{{L}_{j}(x)}{{3}^{j+1}}=\frac{6-x}{8-3x$ |

3 | 1 | 1 | 0 | 2 | 1 | $\sum _{j=0}^{\infty}}\frac{{L}_{j}}{{3}^{j+1}}=1$ |

8 | x | 1 | 0 | 2 | x | $\sum _{j=0}^{\infty}}\frac{{L}_{j}(x)}{{8}^{j+1}}=\frac{16-x}{63-8x$ |

8 | 1 | 1 | 0 | 2 | 1 | $\sum _{j=0}^{\infty}}\frac{{L}_{j}}{{8}^{j+1}}=\frac{3}{11}=\frac{{L}_{2}}{{L}_{5}$ |

$-10$ | x | 1 | 0 | 2 | x | $\sum _{j=0}^{\infty}}\frac{{L}_{j}(x)}{{(-10)}^{j+1}}=\frac{-20-x}{99+10x$ |

$-10$ | 1 | 1 | 0 | 2 | 1 | $\sum _{j=0}^{\infty}}\frac{{L}_{j}}{{(-10)}^{j+1}}=\frac{-21}{109$ |

5 | $2x$ | 1 | 0 | 2 | $2x$ | $\sum _{j=0}^{\infty}}\frac{{Q}_{j}(x)}{{5}^{j+1}}=\frac{5-x}{12-5x$ |

5 | 2 | 1 | 0 | 2 | 2 | $\sum _{j=0}^{\infty}}\frac{{Q}_{j}}{{5}^{j+1}}=\frac{4}{7$ |

3 | 1 | $2y$ | 0 | 2 | 1 | $\sum _{s=0}^{\infty}}\frac{{j}_{s}(y)}{{3}^{s+1}}=\frac{5}{6-2y$ |

3 | 1 | 2 | 0 | 2 | 1 | $\sum _{s=0}^{\infty}}\frac{{j}_{s}}{{3}^{s+1}}=\frac{5}{4$ |

4 | 2 | 2 | $-1$ | $1/4$ | $1/4$ | $\sum _{j=0}^{\infty}}\frac{{F}_{j}^{2}}{{4}^{j}}=\frac{12}{25$ |

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**MDPI and ACS Style**

Kızılateş, C.; Çekim, B.; Tuğlu, N.; Kim, T.
New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers. *Symmetry* **2019**, *11*, 264.
https://doi.org/10.3390/sym11020264

**AMA Style**

Kızılateş C, Çekim B, Tuğlu N, Kim T.
New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers. *Symmetry*. 2019; 11(2):264.
https://doi.org/10.3390/sym11020264

**Chicago/Turabian Style**

Kızılateş, Can, Bayram Çekim, Naim Tuğlu, and Taekyun Kim.
2019. "New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers" *Symmetry* 11, no. 2: 264.
https://doi.org/10.3390/sym11020264