Show that recursion

Question

Hi I have the following problem:

Let $$c_{v,\mu}=\int_a^bw_v(t)(b-t)^{\mu}dt\:\:\:\:\:\:\:,w_v(t)=\prod_{l=0}^{v-1}(t-t_{j-l})$$ with $v=0,...$ and $\mu=0,...$

Show that $$c_{v+1,\mu}=(b-t_{j-v})c_{v,\mu}-c_{v,\mu+1}$$

I already tried to calculate the integral, but I would have to use the integration by parts far too often. I think there is an easier solution to that problem. Can someone help me here please? Thanks in advance.

$w_v(t)=\prod_{t=0}^{v-1}(t-t_{j-l})$ what are $j$ and $l$ ... and $t_{?}$ ? — Donald Splutterwit, Jul 23 '17 at 21:59
Do you mean $$w_v(t)=\prod_{\color{red}{l}=0}^{v-1}(t-t_{j-l})$$? — ChargeShivers, Jul 23 '17 at 22:24

score 2 · Accepted Answer · answered Jul 23 '17 at 22:34

Notice that $$ c_{\nu+1,\mu} = \int_a^b (t-t_{j-\nu})\; w_\nu(t)\;(b-t)^\mu \;\mathrm{d}t$$ and $$ c_{\nu,\mu+1} = \int_a^b (b-t)\; w_\nu(t)\;(b-t)^\mu \;\mathrm{d}t$$ Adding above equations, $$ c_{\nu+1,\mu} + c_{\nu,\mu+1} = \int_a^b (b-t_{j-\nu})\; w_\nu(t)\;(b-t)^\mu \;\mathrm{d}t = (b-t_{j-\nu}) \;c_{\mu,\nu}$$

Show that recursion

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