  In this work, we propose two information generating functions: general
weighted information and relative information generating functions, and study
their properties. { It is shown that the general weighted information
generating function (GWIGF) is shift-dependent and can be expressed in terms of
the weighted Shannon entropy. The GWIGF of a transformed random variable has
been obtained in terms of the GWIGF of a known distribution. Several bounds of
the GWIGF have been proposed. We have obtained sufficient conditions under
which the GWIGFs of two distributions are comparable. Further, we have
established a connection between the weighted varentropy and varentropy with
proposed GWIGF. An upper bound for GWIGF of the sum of two independent random
variables is derived. The effect of general weighted relative information
generating function (GWRIGF) for two transformed random variables under
strictly monotone functions has been studied. } Further, these information
generating functions are studied for escort, generalized escort and mixture
distributions. {Specially, we propose weighted $\beta$-cross informational
energy and establish a close connection with GWIGF for escort distribution.}
The residual versions of the newly proposed generating functions are considered
and several similar properties have been explored. A non-parametric estimator
of the residual general weighted information generating function is proposed. A
simulated data set and two real data sets are considered for the purpose of
illustration. { Finally, we have compared the non-parametric approach with a
parametric approach in terms of the absolute bias and mean squared error
values.}
