deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞< x <∞. ...........(11.3.13) 8. When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN(x). (a) Use Equation (11.3.13) to show that HN(x) satisﬁes (e−x2H′ N)′+2Ne−x2HN =0. ( [Hint: Replace α with N in Equation (11.3.13) and multiply the resulting equation by e−x2.]

deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞< x <∞. ...........(11.3.13) 8. When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN(x). (a) Use Equation (11.3.13) to show that HN(x) satisﬁes (e−x2H′ N)′+2Ne−x2HN =0. ( [Hint: Replace α with N in Equation (11.3.13) and multiply the resulting equation by e−x2.]

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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Question

deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞< x <∞. ...........(11.3.13)

8. When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN(x). (a) Use Equation (11.3.13) to show that H_N(x) satisﬁes

(e^−x2H′ N)′+2Ne^−x2HN =0.

( [Hint: Replace α with N in Equation (11.3.13) and multiply the resulting equation by e^−x2.]

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