Theory and Modern Applications
Table 1 Some new and known special polynomials
From: Identities involving 3-variable Hermite polynomials arising from umbral method
S. No. | Parameters | Polynomials | Umbral definition | Generating function | Series definition |
|---|---|---|---|---|---|
I. | q = 1 | \(H_{n}(x,y,z|\beta ,\alpha ;p,1)\) | \({\hat{b}_{y}}^{\beta }{\hat{c}_{z}}^{p}(x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}}e_{(\alpha ,\beta )}(y^{\frac{\alpha }{2}}t)z^{\frac{p}{3}}\mathcal{E}_{p,1}(z^{\frac{1}{3}}t)\) | \(n!\sum_{r=0}^{n}\frac{\Gamma {(p+r+1)}z^{\frac{p+r}{3}}H_{n-r}(x,y|\beta ,\alpha )}{\Gamma (\frac{p+r}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+r)\pi }{3} \vert - \vert \cos (p+r)\pi \vert )\) |
II. | α = 1 | \(H_{n}(x,y,z|\beta ,1;p,q)\) | \({\hat{b}_{y}}^{\beta }{\hat{c}_{z}}^{p}(x+\hat{b}_{y}+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}}e_{(1,\beta )}(y^{\frac{1}{2}}t)z^{\frac{p}{3}}\mathcal{E}_{p,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n}\frac{\Gamma {(p+qr+1)}z^{\frac{p+qr}{3}}H_{n-r}(x,y|\beta ,1)}{\Gamma (\frac{p+qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+qr)\pi }{3} \vert - \vert \cos (p+qr)\pi \vert )\) |
III. | α = 1; q = 1 | \(H_{n}(x,y,z|\beta ,1;p,1)\) | \({\hat{b}_{y}}^{\beta }{\hat{c}_{z}}^{p}(x+\hat{b}_{y}+\hat{c}_{z})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}} e_{(1,\beta )}(y^{\frac{1}{2}}t)z^{\frac{p}{3}}\mathcal{E}_{p,1}(z^{\frac{1}{3}}t)\) | \(n!\sum_{r=0}^{n}\frac{\Gamma {(p+r+1)}z^{\frac{p+r}{3}}H_{n-r}(x,y|\beta ,1)}{\Gamma (\frac{p+r}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+r)\pi }{3} \vert - \vert \cos (p+r)\pi \vert )\) |
IV. | p = 0 | \(H_{n}(x,y,z|\beta ,\alpha ;-,q)\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}} e_{(\alpha ,\beta )}(y^{\frac{\alpha }{2}}t)\mathcal{E}_{0,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(qr+1)}z^{\frac{qr}{3}}H_{n-r}(x,y|\beta ,\alpha )}{\Gamma (\frac{qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(qr)\pi }{3} \vert - \vert \cos (qr)\pi \vert )\) |
V. | p = 0; q = 1 | \(H_{n}(x,y,z|\beta ,\alpha ;-,1)\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}} e_{(\alpha ,\beta )}(y^{\frac{\alpha }{2}}t)e^{zt^{3}}\) | \(n!\sum_{r=0}^{\frac{n}{3}} \frac{z^{r}H_{n-3r}(x,y|\beta ,\alpha )}{r!(n-3r)!}\) |
VI. | α = 1; p = 0 | \(H_{n}(x,y,z|\beta ,1;-,q)\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y}+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}} e_{(1,\beta )}(y^{\frac{1}{2}}t)\mathcal{E}_{0,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(qr+1)}z^{\frac{qr}{3}}H_{n-r}(x,y|\beta ,1)}{\Gamma (\frac{qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(qr)\pi }{3} \vert - \vert \cos (qr)\pi \vert )\) |
VII. | α = 1; p = 0; q = 1 | \(H_{n}(x,y,z|\beta ,1;-,1)\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y}+{\hat{c}_{z}})^{n}\phi _{0}\psi _{0}\) | \(e^{xt}y^{\frac{\beta }{2}}e_{(1,\beta )}(y^{\frac{1}{2}}t) e^{zt^{3}}\) | \(n!\sum_{r=0}^{[\frac{n}{3}]} \frac{z^{r}H_{n-3r}(x,y|\beta ;1)}{r!(n-3r)!}\) |
VIII. | β = 0 | \(H_{n}(x,y,z|-,\alpha ;p,q)\) | \({\hat{c}_{z}}^{p}(x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt} e_{(\alpha ,0)}(y^{\frac{\alpha }{2}}t)z^{\frac{p}{3}}\mathcal{E}_{p,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(p+qr+1)}z^{\frac{p+qr}{3}}H_{n-r}(x,y|-,\alpha )}{\Gamma (\frac{p+qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+qr)\pi }{3} \vert - \vert \cos (p+qr)\pi \vert )\) |
IX. | β = 0; q = 1 | \(H_{n}(x,y,z|-,\alpha ;p,1)\) | \({\hat{c}_{z}}^{p}(x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}})^{n}\phi _{0}\psi _{0}\) | \(e^{xt} e_{(\alpha ,0)}(y^{\frac{\alpha }{2}}t)z^{\frac{p}{3}}\mathcal{E}_{p,1}(z^{\frac{1}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(p+r+1)}z^{\frac{p+r}{3}}H_{n-r}(x,y|-,\alpha )}{\Gamma (\frac{p+r}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+r)\pi }{3} \vert - \vert \cos (p+r)\pi \vert )\) |
X. | α = 1; β = 0 | \(H_{n}(x,y,z|-,1;p,q)\) | \({\hat{c}_{z}}^{p}(x+\hat{b}_{y}+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt+yt^{2}}z^{\frac{p}{3}} \mathcal{E}_{p,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(p+qr+1)}z^{\frac{p+qr}{3}}H_{n-r}(x,y)}{\Gamma (\frac{p+qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+qr)\pi }{3} \vert - \vert \cos (p+qr)\pi \vert )\) |
XI. | α = 1; β = 0; q = 1 | \(H_{n}(x,y,z|-,1;p,1)\) | \({\hat{c}_{z}}^{p}(x+\hat{b}_{y}+\hat{c}_{z})^{n}\phi _{0}\psi _{0}\) | \(e^{xt+yt^{2}}z^{\frac{p}{3}}\mathcal{E}_{p,1}(z^{\frac{1}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(p+r+1)}z^{\frac{p+r}{3}}H_{n-r}(x,y)}{\Gamma (\frac{p+r}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(p+r)\pi }{3} \vert - \vert \cos (p+r)\pi \vert )\) |
XII. | β = 0; p = 0 | \(H_{n}(x,y,z|-,\alpha ;-,q)\) | \((x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt} e_{(\alpha ,0)}(y^{\frac{\alpha }{2}}t)\mathcal{E}_{0,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(qr+1)}z^{\frac{qr}{3}}H_{n-r}(x,y|-,\alpha )}{\Gamma (\frac{qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(qr)\pi }{3} \vert - \vert \cos (qr)\pi \vert )\) |
XIII. | β = 0; p = 0; q = 1 | \(H_{n}(x,y,z|-,\alpha ;-,1)\) | \((x+\hat{b}_{y}^{\alpha }+{\hat{c}_{z}})^{n}\phi _{0}\psi _{0}\) | \(e^{xt} e_{(\alpha ,0)}(y^{\frac{\alpha }{2}}t)e^{zt^{3}}\) | \(n!\sum_{r=0}^{[\frac{n}{3}]} \frac{z^{r}H_{n-3r}(x,y|-,\alpha )}{r!(n-3r)!}\) |
XIV. | α = 1; β = 0; p = 0 | \(H_{n}(x,y,z|-,1;-,q)\) | \((x+\hat{b}_{y}+{\hat{c}_{z}}^{q})^{n}\phi _{0}\psi _{0}\) | \(e^{xt+yt^{2}} \mathcal{E}_{0,q}(z^{\frac{q}{3}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(qr+1)}z^{\frac{qr}{3}}H_{n-r}(x,y)}{\Gamma (\frac{qr}{3}+1)r!(n-r)!}\times ( \vert 2\cos \frac{(qr)\pi }{3} \vert - \vert \cos (qr)\pi \vert )\) |
XV. | α = 1; β = 0; p = 0; q = 1 | \(H_{n}(x,y,z)\) | \((x+\hat{b}_{y}+\hat{c}_{z})^{n}\phi _{0}\psi _{0}\) | \(e^{(xt+yt^{2}+zt^{3})}\) [6] | \(n!\sum_{r=0}^{[\frac{n}{3}]}\frac{z^{r} H_{n-3r}(x,y)}{r!(n-3r)!}\) [6] |
XVI. | p = 0; q = 0; z = 0 | \(H_{n}(x,y|\beta ,\alpha )\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y}^{\alpha })^{n}\phi _{0}\) [8] | \(e^{xt} y^{\frac{\beta }{2}}e_{(\alpha ,\beta )}(y^{\frac{\alpha }{2}}t)\) [8] | \(n!\sum_{r=0}^{n} \frac{\Gamma {(\alpha r+\beta +1)}x^{n-r}y^{\frac{\alpha r+\beta }{2}}}{\Gamma {(\frac{\alpha r+\beta }{2}+1)}r!(n-r)!}\times \vert (\cos \frac{\alpha r+\beta }{2}\pi ) \vert \) |
XVII. | p = 0; q = 0; α = 1; z = 0 | \(H_{n}(x,y|\beta ,1)\) | \({\hat{b}_{y}}^{\beta }(x+\hat{b}_{y})^{n}\phi _{0}\) | \(e^{xt} y^{\frac{\beta }{2}}e_{(1,\beta )}(y^{\frac{1}{2}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {( r+\beta +1)}x^{n-r}y^{\frac{r+\beta }{2}}}{\Gamma {(\frac{ r+\beta }{2}+1)}r!(n-r)!}\times \vert (\cos \frac{r+\beta }{2}\pi ) \vert \) |
XVIII. | β = 0; p = 0; q = 0; z = 0 | \(H_{n}(x,y|-,\alpha )\) | \((x+\hat{b}_{y}^{\alpha })^{n}\phi _{0}\) | \(e^{xt} e_{(\alpha ,0)}(y^{\frac{\alpha }{2}}t)\) | \(n!\sum_{r=0}^{n} \frac{\Gamma {(\alpha r+1)}x^{n-r}y^{\frac{\alpha r}{2}}}{\Gamma {(\frac{\alpha r}{2}+1)}r!(n-r)!}\times \vert (\cos \frac{\alpha r}{2}\pi ) \vert \) |
XIX. | β = 0; p = 0; α = 1; q = 0; z = 0 | \(H_{n}(x,y)\) | \((x+\hat{b}_{y})^{n}\phi _{0}\) [8] | \(e^{xt+yt^{2}}\) [2] | \(n!\sum_{r=0}^{\frac{n}{2}} \frac{x^{n-2r}y^{r}}{r!(n-2r)!}\) [2] |