A Discrete Wavelet Transform is usually designed with one mother and father wavelet which are generated by a sequence of convolutions of discrete FIR filters.
The mother wavelet has $$\int \psi(t)dt = 0$$ And the father wavelet has $$\int\phi(t)dt=1$$
Then we have dual functions.
Mother first: $$\int \tilde \psi(t)dt = 0$$ And father: $$\int \tilde \phi(t)dt = 1$$
which for orthogonal DWT $\phi = \tilde \phi$, $\psi = \tilde \psi$
but for biorthogonal DWT $\phi \neq \tilde \phi$, $\psi \neq \tilde \psi$
However, there is only one $\psi$. How could we expand this to have several different $\psi$:s.
For example a local differential basis:
$\psi_1$ approximates a derivative whereas.
$\psi_2$ approximates a second derivative.
Note that some special cases of this already exist in some sense, for example many complex Wavelet Transforms have complex functions as mother or father wavelets, maybe the most famous one the continous complex Morlet.
But how to make it discrete and expand to allow for more number of differentiations in a local scale space?
Yes, there are. The first systematic study of examples are the "batman" wavelets of Belogay/Wang (1999): "Construction of compactly supported symmetric scaling functions". One can construct them for any number of bands, making the scaling functions smooth is an optimization task.
For instance, with a 3-channel decomposition with one scaling and two wavelet functions one can get the orthogonal and symmetric wavelets
where the first wavelet has roughly the $[-1,3,-3,1]$ structure of a third order difference quotient and the second less clearly the structure of a fourth order difference quotient. The scaling sequence has approximation order 5, that is, polynomial trends up to degree 4 are mostly passed through the low-pass band.
Likewise symmetric and orthogonal is the 4-channel wavelet basis
which gave some nice results in experimental image compression