Theory and Modern Applications
Table 2 Index and argument duplication formulas
From: Identities involving 3-variable Hermite polynomials arising from umbral method
S. No. | Polynomials | Index duplication formula | Argument duplication formula |
|---|---|---|---|
I. | \(H_{n}(x,y,z|\beta ,\alpha ;p,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s)+\beta ,\alpha ;p+n-r,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u)+\beta ,\alpha ;s-r+p,1 )\) |
II. | \(H_{n}(x,y,z|\beta ,1;p,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s+\beta ,1;q(n-r)+p,q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u+\beta ,1; q(s-r)+p,q )\) |
III. | \(H_{n}(x,y,z|\beta ,1;p,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s+\beta ,1;n-r+p,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u+\beta ,1; s-r+p,1 )\) |
IV. | \(H_{n}(x,y,z|\beta ,\alpha ;-,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s)+\beta ,\alpha ;q(n-r),q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u)+\beta ,\alpha ;q(s-r),q )\) |
V. | \(H_{n}(x,y,z|\beta ,\alpha ;-,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s)+\beta ,\alpha ;n-r,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{s-r-u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u)+\beta ,\alpha ; s-r,1 )\) |
VI. | \(H_{n}(x,y,z|\beta ,1;-,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s+\beta ,1;q(n-r),q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u+\beta ,1; q(s-r),q )\) |
VII | \(H_{n}(x,y,z|\beta ,1;-,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s+\beta ,1;n-r,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u+\beta ,1;s-r,1 )\) |
VIII. | \(H_{n}(x,y,z|-,\alpha ;p,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s),\alpha ;q(n-r)+p,q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u),\alpha ; q(s-r)+p,q )\) |
IX. | \(H_{n}(x,y,z|-,\alpha ;p,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s),\alpha ;n-r+p,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u),\alpha ; s-r+p,1 )\) |
X. | \(H_{n}(x,y,z|-,1;p,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s,1;q(n-r)+p,q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u,1;q(s-r)+p,q )\) |
XI. | \(H_{n}(x,y,z|-,1;p,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s,1;n-r+p,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u,1; s-r+p,1 )\) |
XII. | \(H_{n}(x,y,z|-,\alpha ;-,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s),\alpha ;q(n-r),q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u),\alpha ;q(s-r),q )\) |
XIII. | \(H_{n}(x,y,z|-,\alpha ;-,1)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|\alpha (r-s),\alpha ;n-r,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|\alpha (r-u),\alpha ;s-r,1 )\) |
XIV. | \(H_{n}(x,y,z|-,1;-,q)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s,1;q(n-r),q)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u,1;q(s-r),q )\) |
XV. | \(H_{n}(x,y,z)\) | \(\sum_{r=0}^{n}\sum_{s=0}^{r}\binom{n}{r}\binom{r}{s}x^{s}H_{n}(x,y,z|r-s,1;n-r,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\sum_{u=0}^{r}\binom{n}{s}\binom{s}{r}\binom{r}{u}x^{u}\frac{1}{2^{s-u}}\times H_{n-s} (x,\frac{y}{2},\frac{z}{2}|r-u,1;s-r,1 )\) |
XVI. | \(H_{n}(x,y|\beta ,\alpha )\) | \(\sum_{r=0}^{n}\binom{n}{r}x^{r}H_{n}(x,y|\alpha (n-r)+\beta ,\alpha )\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\binom{n}{s}\binom{s}{r}x^{r}\frac{1}{2^{s-r}}\times H_{n-s} (x,\frac{y}{2}|\alpha (s-r)+\beta ,\alpha )\) |
XVII. | \(H_{n}(x,y|\beta ,1)\) | \(\sum_{r=0}^{n}\binom{n}{r}x^{r}H_{n}(x,y|n-r+\beta ,1)\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\binom{n}{s}\binom{s}{r}x^{r}\frac{1}{2^{s-r}}\times H_{n-s} (x,\frac{y}{2}|s-r+\beta ,1 )\) |
XVIII. | \(H_{n}(x,y|-,\alpha )\) | \(\sum_{r=0}^{n}\binom{n}{r}x^{r}H_{n}(x,y|\alpha (n-r),\alpha )\) | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\binom{n}{s}\binom{s}{r}x^{r}\frac{1}{2^{s-r}}\times H_{n-s} (x,\frac{y}{2}|\alpha (s-r),\alpha )\) |
XIX. | \(H_{n}(x,y)\) | \(\sum_{r=0}^{n}\binom{n}{r}x^{n-r}H_{n}(x,y|r)\) [8] | \(\sum_{s=0}^{n}\sum_{r=0}^{s}\binom{n}{s}\binom{s}{r}x^{r}\frac{1}{2^{s-r}}\times H_{n-s} (x,\frac{y}{2}|s-r )\) [8] |