Mehler
In mathematics, particularly in the field of special functions and orthogonal polynomials, the name "Mehler" is associated with several concepts and formulas, most notably related to Hermite polynomials. These associations often stem from the work of German mathematician Friedrich Gustav Christoph Mehler.
Mehler's Formula (for Hermite Polynomials):
Mehler's formula, also known as Mehler's kernel, provides a closed-form expression for the sum of the product of two Hermite polynomials, weighted by an exponential factor. It is given by:
∑n=0∞ Hn(x) Hn(y) tn / n! = (1 - t2)-1/2 exp(2xyt - (x2 + y2)t2) / (1 - t2)
where:
- Hn(x) represents the nth-degree Hermite polynomial.
- x and y are variables.
- t is a real number such that |t| < 1.
This formula is crucial in the analysis and applications of Hermite polynomials, particularly in quantum mechanics and probability theory. It allows for the efficient calculation of certain sums involving these polynomials.
Mehler-Heine Formula:
Another related result, known as the Mehler-Heine formula (or theorems), describes the asymptotic behavior of orthogonal polynomials, including Hermite polynomials, as the degree of the polynomial tends to infinity. These formulas generally state that suitably scaled orthogonal polynomials converge to certain elementary functions, offering valuable insights into their behavior for large degrees.
Other Associations:
While the Hermite polynomial formulas are the most common association, Mehler's name might also appear in other contexts related to mathematical analysis and physics, although these are less frequent. Any such usage would require clarification based on the specific context.