## Integral identity for Legendre polynomials

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial? ```$$ 
\int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1}
$$```

### Notes & Background

• A variant of this appears in, for instance, Erdelyi et al "Higher transcendental functions" 10.10(49), but with nothing in the way of explanation.

• This comes up in harmonic analysis on $U(3)$, when comparing Gelfand-Tseltin bases associated to different choices of nested sequences $U(3) \supset U(2) \supset U(1)$.

• Eventually, I'll be looking for a $q$-analogue, related to harmonic analysis on $U_q(3)$, so a proof that will transport well would be my true desire.

-

2
By means of power series, this transforms into something like $\int\limits_0^1\dfrac{1}{\sqrt{1-2\left(2x^2-1\right)t+t^2}}dx=\mathrm{arctan}t$ (note that I have changed your $t$ into an $x$). Maybe it's easier this way? – darij grinberg Apr 21 2011 at 9:30

2
$$\int_0^1 \frac 1 {\sqrt{1-2(2x^2-1)t^2+t^4}}dx = \frac {\arctan t} t$$ might work even better. – thei Apr 21 2011 at 10:30

## 1 Answer

Nice idea. As far as I'm concerned, the above comments are "answers", since they check out. I might as well record the details: ```$$
\int_0^1 \frac{dx}{\sqrt{1-2(2x^2-1)t + t^2}} 
= \frac{1}{2\sqrt{t}} \int_0^1 \frac{dx}{\sqrt{\frac{(1+t)^2}{4t} -x^2}} = \frac{1}{2\sqrt{t}}\arcsin\left(\frac{\scriptstyle 2\sqrt{t}}{\scriptstyle1+t}\right).
$$``` The half-angle formula for $\sin$ reduces this to $$\frac{1}{\sqrt{t}} \arcsin \sqrt{\frac{t}{1+t}} = \frac{1}{\sqrt{t}} \arctan \sqrt{t},$$ which has the desired power series expansion.

Thanks.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8553212881088257, "perplexity_flag": "middle"}