Start with the Chebyshev polynomials of the first kind, omitting even values:

T₁(x)=x
T₃(x)=4x³-3x
T₅(x)=16x⁵-20x³+5x
T₇(x)=64x⁷ -112x⁵ +56x³ -7x
....etc.

Change all minus signs to plusses, and replace each exponent, m, by (m-1)/2:

T₁(x)=1
T₃(x)=4x+3
T₅(x)=16x² +20x+5
T₇(x)=64x³ +112x² +56x+7
....etc., to produce (omitting (x) for clarity of exposition) the new polynomials, P₁, P₂, P₃, P₄...P_n.

Similarly, take the Chebyshev polynomials of the second kind, this time omitting odd values :

U₀(x) = 1
U₂(x) = (4x² -1)
U₄(x) = (16x⁴ -12x² +1)
U₆(x) = (64x⁶ -80x⁴ +24x² -1)
....etc.

Change all minus signs to plusses, and replace each exponent, m, by (m/2):

U₀(x) = 1
U₂(x) = (4x+1)
U₄(x) = (16x² +12x+1)
U₆(x) = (64x³ +80x² +24x+1)
....etc., to produce (omitting (x) for clarity of exposition) the new polynomials, Q₁, Q₂, Q₃, Q₄...Q_n.

Amazingly, (P_n)² *x+1 = (Q_n)² *(x+1)

Prove it!