Let Z−=∖{−1,−2,…}. In 2023, Qi and Lim gave two claims for summing the infinite series k=1∑∞(k2k)α+k1(4±1)k,α∈C∖Z−.
Nearby in the stack
In present paper, the authors establish several sum functions of the infinite and finite series
k=1∑∞(k2k)α+k1(4z)kandk=1∑n(k2k)α+k1(4z)k
for
α∈C∖Z−
and
n∈N={1,2,…}
in terms of the Gauss hypergeometric functions
2F1
and the generalized hypergeometric functions
3F2
for
α∈C∖Z−
and
n∈N
. In light of the Euler integral representation of the Gauss hypergeometric function
2F1
, the author present several closed forms of two Gauss hypergeometric functions
2F1
, two generalized hypergeometric functions
3F2
, and the classical incomplete beta functions
Bz(21,21+n)
and
Bz(21,1+n)
. With the help of the Euler hypergeometric transform, the authors derive closed forms of five Gauss hypergeometric functions. In addition, the authors also obtain a closed form of the differential operator