For linear recurrences U and V, a polynomial P turns d P(n) U(m)/V(n) into a recurrence and V(n)/P(n) linear, for m not equal n as multi-recurrence and for m=n as linear.
Andr´ e-Jeannin,Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart.29(4) (1991), 364-366
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The quotient problem for linear recurrence sequences
For linear recurrences U and V, a polynomial P turns d P(n) U(m)/V(n) into a recurrence and V(n)/P(n) linear, for m not equal n as multi-recurrence and for m=n as linear.