Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a
q
q
-analogue of the Euler constant and proved the irrationality of certain numbers involving
q
q
-Euler constant. In this paper, we improve their results and prove the linear independence result involving
q
q
-analogue of the Euler constant. Further, we derive the closed-form of a
q
q
-analogue of the
k
k
-th Stieltjes constant
γ
k
(
q
)
\gamma _k(q)
. These constants are the coefficients in the Laurent series expansion of a
q
q
-analogue of the Riemann zeta function around
s
=
1
s=1
. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series
∑
n
≥
1
σ
1
(
n
)
/
t
n
\sum _{n\geq 1}{\sigma _1(n)}/{t^n}
for any integer
t
>
1
t > 1
. Finally, we study the transcendence nature of some infinite series involving
γ
1
(
2
)
\gamma _1(2)
.