Is the following polynomial positive: T_k(t)=(t/2)^p sum_(j=0)^k((−t^2/4)j Gamma(p+1))/(j! Gamma(p+j+1)).

Macioccujx 2022-07-20 Answered
Is the following polynomial positive:
T k ( t ) = ( t 2 ) p j = 0 k ( t 2 4 ) j Γ ( p + 1 ) j ! Γ ( p + j + 1 ) .
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Answers (1)

kuglatid4
Answered 2022-07-21 Author has 12 answers
Consider series
T ( t ) = ( t 2 ) p j = 0 ( t 2 4 ) j Γ ( p + 1 ) j ! Γ ( j + p + 1 )
This is related to the series representation of the Bessel function of order p of the first kind. Indeed
T ( t ) = Γ ( p + 1 ) j = 0 ( 1 ) j j ! Γ ( j + p + 1 ) ( t 2 ) 2 j + p = Γ ( p + 1 ) J p ( t )
Since this series converges, then
(1) lim k T k ( t ) = Γ ( p + 1 ) J p ( t )
It is known that Bessel functions of the first kind take negative and positive values infinitely many times on ( 0 , + ). Hence we may consider t 0 such that Γ ( p + 1 ) J p ( t 0 ) < 0. From (1) it follows that for some k 0 we would have
T k ( t 0 ) < 0  for all k > k 0 .

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