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darij grinberg
darij grinberg
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In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the following formula to Mehler (quoting a book by Erdelyi): $$ \sum_{n=0}^{\infty}\frac{r^n H_n(x) H_n(y)}{2^n n!}=(1-r^2)^{-1/2} \exp\bigl\{ x^2-\frac{(x-ry)^2}{(1-r^2)} \bigr\}, $$ where $H_n$ is the $n$th Hermite polynomial.

Question: I do not believe that formula, since the lhs is symmetric in $x,y$ whereas the rhs fails to be symmetric in $x,y$. I am also puzzled since, as said above, this article is a reference material for numerous articles on Laguerre and Hermite polynomials.

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the following formula to Mehler (quoting a book by Erdelyi): $$ \sum_{n=0}^{\infty}\frac{r^n H_n(x) H_n(y)}{2^n n!}=(1-r^2)^{-1/2} \exp\bigl\{ x^2-\frac{(x-ry)^2}{(1-r^2)} \bigr\}, $$ where $H_n$ is the $n$th Hermite polynomial.

Question: I do not believe that formula, since the lhs is symmetric in $x,y$ whereas the rhs fails to be symmetric in $x,y$. I am also puzzled since, as said above, this article is a reference material for numerous articles on Laguerre and Hermite polynomials.

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the following formula to Mehler (quoting a book by Erdelyi): $$ \sum_{n=0}^{\infty}\frac{r^n H_n(x) H_n(y)}{2^n n!}=(1-r^2)^{-1/2} \exp\bigl\{ x^2-\frac{(x-ry)^2}{(1-r^2)} \bigr\}, $$ where $H_n$ is the $n$th Hermite polynomial.

Question: I do not believe that formula, since the lhs is symmetric in $x,y$ whereas the rhs fails to be symmetric in $x,y$. I am also puzzled since, as said above, this article is a reference material for numerous articles on Laguerre and Hermite polynomials.

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Bazin
Bazin
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On Mehler's formula for Hermite polynomials

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the following formula to Mehler (quoting a book by Erdelyi): $$ \sum_{n=0}^{\infty}\frac{r^n H_n(x) H_n(y)}{2^n n!}=(1-r^2)^{-1/2} \exp\bigl\{ x^2-\frac{(x-ry)^2}{(1-r^2)} \bigr\}, $$ where $H_n$ is the $n$th Hermite polynomial.

Question: I do not believe that formula, since the lhs is symmetric in $x,y$ whereas the rhs fails to be symmetric in $x,y$. I am also puzzled since, as said above, this article is a reference material for numerous articles on Laguerre and Hermite polynomials.