| We define k-interpolated sequences by a family of divide-and-conquer recurrence equations. These sequences are k-regular after a segment of initial values. A k-regular sequence is computed with an auxiliary vector. For k-interpolated sequences, this vector is embedded in the sequence itself.;The generating function of a k-interpolated sequence satisfies a functional equation of Mahlerian type. We show when such a functional equation is solved by the generating function of a k-interpolated sequence, and how the parameters of that sequence are determined. Furthermore, a conjugate k-interpolated sequence is defined so that the generating function is extended to the complex plane less the unit circle. Finally, we show that unrolling the recursion in the functional equation yields a sum convergent away from the unit circle.;We define a space of testing functions so that the extended generating functions act as functionals. The range of the Mellin transform on these testing functions is a space of analytic functions. The dual Mellin transform on the space of functionals has an operational calculus similar to the Mellin transform. In some cases, it takes multiplication to convolution, and Laurent polynomials are transformed into interpolation operators. |
