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 <a <b <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.