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Keywords:
n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences
n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences
Summary:
Suppose that the function $q(t)$ in the differential equation (1) $y^{\prime \prime }+q(t)y=0 $ is decreasing on $(b,\infty )$ where $b \ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1(t),\;y_2(t)$ such that the $n$-th derivative ($n\ge 1$) of the function $p(t)= y_1^2(t) +y_2^2(t)$ has the sign $(- 1)^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.
References:
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