Question

3. The generalised Laguerre differential equation is $xy''(x) + (\alpha + 1 - x)y'(x) + ny = 0$. (a) Substitute a solution of the form $y(x) = \sum_{k=0}^{\infty} a_k x^k$ into the equation and hence derive a recurrence relation for the coefficients $a_k$. Deduce that for positive integer $n$, $y(x)$ is a polynomial. (b) The Laguerre polynomials are given by $L_n^{(\alpha)}(x) = \sum_{k=0}^n (-1)^k \binom{n+\alpha}{n-k} \frac{x^k}{k!}$ Verify that these satisfy the recurrence relation you found in part (a).

          3. The generalised Laguerre differential equation is
$xy''(x) + (\alpha + 1 - x)y'(x) + ny = 0$.
(a) Substitute a solution of the form
$y(x) = \sum_{k=0}^{\infty} a_k x^k$
into the equation and hence derive a recurrence relation for the coefficients $a_k$.
Deduce that for positive integer $n$, $y(x)$ is a polynomial.
(b) The Laguerre polynomials are given by
$L_n^{(\alpha)}(x) = \sum_{k=0}^n (-1)^k \binom{n+\alpha}{n-k} \frac{x^k}{k!}$ 
Verify that these satisfy the recurrence relation you found in part (a).

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3. The generalised Laguerre differential equation is
xy”(x) + (α + 1 - x)y'(x) + ny = 0.
(a) Substitute a solution of the form
y(x) = ∑k=0^∞ ak x^k
into the equation and hence derive a recurrence relation for the coefficients ak.
Deduce that for positive integer n, y(x) is a polynomial.
(b) The Laguerre polynomials are given by
Ln^(α)(x) = ∑k=0^n (-1)^k n+αn-k(x^k)/(k!)
Verify that these satisfy the recurrence relation you found in part (a).

Added by Belen D.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Suzanne W.

Step 1

The term with x^k is: k(k-1)akx^k The term with x^(k-1) is: (a+1)akx^(k-1) The term with x^k-2 is: -akx^(k-2) So, the equation becomes: k(k-1)akx^k + (a+1)akx^(k-1) - akx^(k-2) + ny = 0 Now, let's collect the terms with the same power of x: x^k: k(k-1)ak Show more…

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The generalized Laguerre differential equation is xy''(x) + (a+1-xy(x)) + ny = 0. (a) Substitute a solution of the form y(x) = Î£akx^k into the equation and hence derive a recurrence relation for the coefficients ak. Deduce that for positive integer n, y(x) is a polynomial. [5] (b) The Laguerre polynomials are given by L_n(x) = Î£akx^k Verify that these satisfy the recurrence relation you found in part (a). [3]

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