|
|
| Line 4: |
Line 4: |
| | | | |
| | =Properties= | | =Properties= |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| |
| − | <strong>Theorem:</strong> The following formula holds:
| |
| − | $$n^2(2n-3)Z_n(x)-(2n-1)[3n^2-6n+2-2(2n-3)x]Z_{n-1}(x)+(2n-3)[3n^2-6n+2+2(2n-1)x]Z_{n-2}(x)-(2n-1)(n-2)^2Z_{n-3}(x)=0.$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − |
| |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| |
| − | <strong>Theorem:</strong> The following formula holds:
| |
| − | $$xZ_n'(x)-nZ_n(x)=-nZ_{n-1}(x)-xZ_{n-1}'(x).$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − |
| |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| |
| − | <strong>Theorem:</strong> The following formula holds:
| |
| − | $$xZ_n'(x)-nZ_n(x)=-\displaystyle\sum_{k=0}^{n-1} Z_k(x) - 2x\displaystyle\sum_{k=0}^{n-1} Z_k'(x).$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − |
| |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| |
| − | <strong>Theorem:</strong> The following formula holds:
| |
| − | $$xZ_n'(x)-nZ_n(x)=\displaystyle\sum_{k=0}^{n-1} (-1)^{n-k}(2k+1)Z_k(x).$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − |
| |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| |
| − | <strong>Theorem:</strong> The following formula holds:
| |
| − | $$\dfrac{1}{1-t} {}_1F_1 \left( \dfrac{1}{2} ; 1 ; -\dfrac{4xt}{(1-t)^2} \right) = \displaystyle\sum_{k=0}^{\infty} Z_n(x)t^n,$$
| |
| − | where ${}_1F_1$ denotes the [[generalized hypergeometric function]].
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − |
| |
| | | | |
| | =References= | | =References= |
| − | Rainville "Special Functions"
| + | * {{PaperReference|Some Properties of a certain Set of Polynomials|1933|Harry Bateman}} |
| | | | |
| | {{:Orthogonal polynomials footer}} | | {{:Orthogonal polynomials footer}} |
| | | | |
| | [[Category:SpecialFunction]] | | [[Category:SpecialFunction]] |
