We introduce the notion of Drinfeld modular forms with A-expansions, where instead of the usual Fourier expansion in t n (t being the uniformizer at 'infinity'), parametrized by n ∈ N, we look at expansions in t a , parametrized by a ∈ A = F q [T ]. We construct an infinite family of eigenforms with Aexpansions. Drinfeld modular forms with A-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) the computation of the eigensystems of Drinfeld modular forms with A-expansions; (iii) examples of failure of multiplicity one result, as well as a restrictive multiplicity one result for Drinfeld modular forms with A-expansions; (iv) examples of eigenforms that can be represented as 'non-trivial' products of eigenforms; (v) an extension of a result of Böckle and Pink concerning the Hecke properties of the space of cuspidal modulo double-cuspidal forms for Γ 1 (T ) to the groups GL 2 (F q [T ]) and Γ 0 (T ).

We show that a certain family of the coefficients of a Drinfeld-Goss modular form with certain power eigenvalues for the Hecke operators at degree 1 primes the can be expressed as polynomial multiples of the first possible non-zero coefficient of that form. Along the way, we obtain some interesting combinatorial properties regarding difference operators in finite characteristic.

We prove the equality of several τ -recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an A-expansion for the l th power (1 ≤ l ≤ q) of the deformation of the weight 2 Eisenstein series; relations between Drinfeld modular forms with A-expansions; a new proof of relations between special values of Pellarin L-series.Both s 1 , s 2 converge for (z, t) ∈ Ω × B q . For arithmetic consideration it is more convenient to work with normalizations of s 1 and s 2 , namelybe the ring homomorphism defined by χ t (a) = a(t). If α, β are positive integers, then the Pellarin L-function is defined byPellarin introduced L(χ α t , β) in [13] as a deformation of the Carlitz zeta function and more general Goss L-functions (see [7, Chapter 8]). In addition to Pellarin's original paper, the reader can find information about analytic continuation of Pellarin's L-function in [8] (note that we will only consider values of L(χ α t , β) for positive integers α, β, i.e., values as in Equation (1.1)), and formulas for special values in [15]. In the course of the proof of our main result we will give a new proof of several relations between special values of Pellarin L-functions (see Corollary 3.4).Let τ : C ∞ ((t)) → C ∞ ((t)) be the field automorphism that fixes t and acts as the Frobenius qth power operator on elements of C ∞ . This agrees with the previous definition of τ on C ∞ , so by abuse of notation we will use τ to denote both. If f ∈ C ∞ ((t)), then we will use the notation f (i) for τ i f .For l ∈ N we define the sequence {G l,k } k∈Z by G l,k := G l,k (z, t) = 1 L(χ l t , lq k )

We show that the Drinfeld modular forms with A-expansions that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with A-expansions that remain modular after hyperdifferentiation. In addition, we show that certain Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with A-expansions are DrinfeldPoincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen and van der Put.