The Pfaffian Transformation
نویسندگان
چکیده
We then define the Pfaffian transformation of (a0, a1, a2, ...) to be the sequence of Pfaffians (Pf(A0), P f(A1), P f(A2), ...), where the Pfaffian of a skew-symmetric matrix is the positive or negative square root of its determinant. (For a precise definition of the Pfaffian, see section 2.) The Pfaffian transformation is thus a function from sequences to sequences. We begin by observing the effects of this transformation on some simple, well-known sequences: (1, 2, 3, 4, 5, ...) → (1, 0, 0, 0, 0, ...) (2, 2, 2, 2, 2, ...) → (2, 4, 8, 16, 32, ...) (1, 2, 4, 8, 16, ...) → (1, 1, 1, 1, 1, ...) (1, 1, 2, 3, 5, ...) → (1, 2, 4, 8, 16, ...) Observe that each of these input sequences satisfies a linear recurrence relation and is mapped to a sequence that also satisfies a linear recurrence. Our main result proves that this holds in general; we also give an algorithm to calculate the output recurrence for such an input sequence.
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