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Keywords:
psi function; asymptotic expansion; complete monotonicity
psi function; asymptotic expansion; complete monotonicity
Summary:
Let $p,q\in \mathbb {R}$\ with $p-q\geq 0$, $\sigma = \frac 12 ( p+q-1)$ and $s=\frac 12 ( 1-p+q)$, and let $$ \mathcal {D}_{m} ( x;p,q ) =\mathcal {D}_{0} ( x;p,q ) +\sum _{k=1}^{m}\frac {B_{2k} ( s) }{2k ( x+\sigma ) ^{2k}} , $$ where $$ \mathcal {D}_{0} ( x;p,q ) =\frac {\psi ( x+p ) +\psi ( x+q ) }{2}-\ln ( x+\sigma ) . $$ We establish the asymptotic expansion $$ \mathcal {D}_{0} ( x;p,q ) \sim -\sum _{n=1}^{\infty } \frac {B_{2n} ( s ) }{2n ( x+\sigma ) ^{2n}} \quad \text {as} \^^Mx\rightarrow \infty , $$ where $B_{2n} ( s ) $ stands for the Bernoulli polynomials. Further, we prove that the functions $( -1) ^{m}\mathcal {D}_{m} ( x;p,q )$ and $( -1) ^{m+1}\mathcal {D}_{m} ( x;p,q )$ are completely monotonic in $x$ on $( -\sigma ,\infty )$ for every $m\in \mathbb {N}_{0}$ if and only if $p-q\in [ 0, \tfrac 12 ]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.
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