# Analysis and classic odes – Lower limit of the standard \$ L ^ 2 \$ of a polynomial

Leave $$sigma> 0$$ be arranged even for $$k in mathbb {N} cup {0 }$$, we consider the polynomial
$$begin {equation} varphi_k (x) = sum_ {j = 0} ^ {k} (-1) ^ j {k choose j} b_j , x ^ {2j} x en (-1,1), end {equation}$$
where
$$begin {equation} b_j = frac { Big (k + sigma + frac12 Big) _j} { Big ( frac12 Big) _j} end {equation}$$
and to $$s in mathbb {R}$$, $$(s) _j$$ denotes the symbol of Pochhammer
$$begin {equation} (s) j = { begin {cases} 1 & j = 0 \ s (s + 1) cdots (s + j-1) & j> 0. end {cases}} end {equation}$$
In particular, $$varphi_0 (x) = 1$$.

My question is this.

by $$-1 exists? $$c> 0$$ such that
$$begin {equation} int_a ^ b | varphi_k (x) | ^ 2 dx geq c end {equation}$$
even for any $$k in mathbb {N} cup {0 }$$?

Unless I'm wrong, a direct calculation yields
$$begin {equation} int_a ^ b | varphi_k (x) | ^ 2 dx = sum_ {j = 0} ^ k sum _ { ell = 0} ^ k (-1) ^ {j + ell} {k choose j} {k choose ell} frac {b_j b _ { ell}} {2 (j + ell) +1} , (b ^ {2 (j + ell) +1} -a ^ {2 (j + ell) +1} ). end {equation}$$
But I do not see how I can join this double sum from below.

Observation: I do not know if it is of any use, one can notice that $$varphi_k$$ It is a hypergeometric function of the form. $${} _ {2} F_ {1} (- k, k + sigma + frac12; frac12; x ^ 2)$$ (See https://en.wikipedia.org/wiki/Hypergeometricometric_function).

Edit: The same question is open when $$k$$ It's strange, but this time you consider it.
$$begin {equation} varphi_k (x) = sum_ {j = 0} ^ {k} (-1) ^ j {k choose j} c_j , x ^ {2j + 1}, end {equation}$$
with
$$begin {equation} c_j = frac { Big (k + sigma + frac32 Big) _j} { Big ( frac32 Big) _j}. end {equation}$$
I suspect that the methodology is similar to that of the even case.

Realize: I have asked the question in MSE, where it is subject to a reward: https://math.stackexchange.com/questions/3154394/bounding-a-polynomial-from-below.