  We generalize previous definitions of Tesler matrices to allow negative
matrix entries and negative hook sums. Our main result is an algebraic
interpretation of a certain weighted sum over these matrices, which we call the
Tesler function. Our interpretation uses a new class of symmetric function
specializations which are defined by their values on Macdonald polynomials. As
a result of this interpretation, we obtain a Tesler function expression for the
Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$
is the delta operator introduced by Bergeron, Garsia, Haiman, and Tesler. We
also provide simple formulas for various special cases of Tesler functions
which involve $q,t$-binomial coefficients, ordered set partitions, and parking
functions. These formulas prove two cases of the recent Delta Conjecture posed
by Haglund, Remmel, and the author.
