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I-a-b-0-1-t-a-1-t-b-dt-Note-I-a-b-a-b-1-I-a-1-b-1-show-that-I-a-1-b-I-a-b-1-I-a-b-find-B-a-1-b-1-interms-of-B-a-b-use-I-a-b-I-b-a-and-deduce-I-5-2-3-2-

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Question Number 208908 by alcohol last updated on 26/Jun/24
I(a,b) = ∫_0 ^( 1) t^a (1−t)^b dt Note : I(a,b) = (a/(b+1))I(a−1,b+1) show that • I(a+1, b) + I(a,b+1) = I(a, b) • find B(a+1, b+1) interms of B(a,b) • use I(a,b) = I(b,a) and deduce I((5/2),(3/2))
$${I}\left({a},{b}\right)\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {t}^{{a}} \left(\mathrm{1}−{t}\right)^{{b}} {dt} \\ $$$${Note}\::\:{I}\left({a},{b}\right)\:=\:\frac{{a}}{{b}+\mathrm{1}}{I}\left({a}−\mathrm{1},{b}+\mathrm{1}\right) \\ $$$${show}\:{that} \\ $$$$\bullet\:{I}\left({a}+\mathrm{1},\:{b}\right)\:+\:{I}\left({a},{b}+\mathrm{1}\right)\:=\:{I}\left({a},\:{b}\right) \\ $$$$\bullet\:{find}\:{B}\left({a}+\mathrm{1},\:{b}+\mathrm{1}\right)\:{interms}\:{of}\:{B}\left({a},{b}\right) \\ $$$$\bullet\:{use}\:{I}\left({a},{b}\right)\:=\:{I}\left({b},{a}\right)\:{and}\:{deduce}\:{I}\left(\frac{\mathrm{5}}{\mathrm{2}},\frac{\mathrm{3}}{\mathrm{2}}\right) \\ $$

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