Convolutions of generalized Fibonacci and Lucas sequences

The Riccati eqs. 13, resp. 14, are replaced in the non-degenerate case by eqs. 15, resp. 16. Because the $k-$th power of the (ordinary) generating functions of a sequence generates $k-$fold convolutions of this sequence one obtains the expression eq. 18, resp. eq. 19. For the definition of the first convolutions $U^{(1)}(a,b)$ and $V^{(1)}(a,b)$ see eqs. 17. Eq. 19 simplifies to eq. 20 after use of the recurrence relation eq. 1. In this way the first convolutions can be determined in each case from linear combinations of the two independent original sequences. For $a\, =\, 1 \, =\, b$ these formulae are well-known (see the Introduction after eq. 20). The generalization to arbitrary $k-$fold convolutions is now straightforward.

Lemma 3 (Recurrencce for $k-$fold convolutions):

In the non-degenerate case $a^2\, +\, 4\,b \neq 0$ the recurrence eq. 25, resp. eq. 26, holds for the $k-$fold convolution of generalized Fibonacci, resp. Lucas, sequences.

Proof: This is a direct consequence of the identities for the $k+1$st power of $U(a,b;x)$, resp. $V(a,b;x)$ shown in eq. 23, resp. eq. 24. These identities, in turn, result after induction over $k$ with the basis $(k=1)$ provided by eq. 15, resp. eq. 16. Because of $U^{k+1}(a,b;x)\ =\ \sum_{n=0}^{\infty}\, U_{n}^{(k)}(a,b)\, x^n$, and the same eq. with $U$ replaced by $V$, one arrives at eq. 25, resp. eq. 26.$\square$

Lemma 4 (Recurrence for $k-$fold convolution, degenerate case):

For $b\, =\, -{\frac{a^2}{4}}\, \neq\, 0$ the recurrence formulae for the $k-$fold convolution of the generalzed Fibonacci, resp. Lucas, sequences are those stated in eqs. 33, resp. 34.

Proof: This statement is equivalent to eq. 31, resp. eq. 32 for the powers of the corresponding generating functions. They are deduced from the the second, resp. first, order differential eq. given in eq. 29 which is identical with the $k=1$ assertion. To verify the general $k$ claim eq. 31, resp. eq. 32, one may use $U(a;x)\, =\, U(a,-{\frac{a^2}{4}};x)\, =\, 1/(1-a\, x/2)^2$ from eq. 2, resp. $V(a;x)\, =\, V(a,-{\frac{a^2}{4}};x)\, =\, 1/(1-a\, x/2)$ from eq. 3.$\square$

Lemma 5: The explicit form for the $k-$fold convolution in the degenerate case is given by eq. 35, resp.  36, for the generalized Fibonacci, resp. Lucas, case.

Proof: Iteration of the recurrence eq. 33, resp. eq. 34, with input $U_{n}^{(0)}(a)\, =\, U_{n}(a)\, =\, (a/2)^n$, resp. $V_{n}^{(0)}(a)\, =\, V_{n}(a)\, =\, (a/2)^n$, which originates from the generating functions $U(a,-{\frac{a^2}{4}};x)$, resp. $V(a,-{\frac{a^2}{4}};x)$. $\square$

Proposition 3 (Iteration of recurrence for $k-$fold convolutions; non-degenerate Fibonacci case):

For $a^2\, +\, 4\, b\, \neq\, 0$ the $k-$fold convolution of the generalized Fibonacci sequence $\{U_{n}(a,b)\}$ is expressed as linear combinations of the two independent solutions of recurrence eq. 1 as given in eq. 37. The coefficient polynomials $AU_{k}(a,b;n)$ and $BU_{k}(a,b;n)$ satisfy the mixed recurrence relations eqs. 38 and  39.

Proof: If one considers eq. 37 as ansatz and puts it into the recurrence eq. 25 one finds, after elimination of $U_{n+2}(a,b)$ via its recursion relation and a comparison of the coefficients of the linear independent $U_{n}(a,b)$ and $U_{n-1}(a,b)$ sequences, the mixed recurrence relations for $AU_{k}(a,b;n)$ and $BU_{k}(a,b;n)$. The inputs $AU_{0}(a,b;n)\, =\, 1$ and $BU_{0}(a,b;n)\, =\, 2$ are necessary in order that for $k=1$ eq. 37 coincides with eq. 18. With these inputs and the mixed recurrence one proves, by induction over $k$, that $AU_{k}(a,b;n)$ and $BU_{k}(a,b;n)$ are polynomials in $n$ of degree $k$, provided $a$ and $b$ are fixed with $b\, \neq\, -a^2/4$, $b\,\neq\, 0$ and $a \, \neq\, 0$. $\square$

Note 2: For integers $a$ and $b\neq\, -a^2/4$ the coefficients of the polynomials $AU_{n}(a,b;x)$ and $BU_{n}(a,b;x)$ furnish two lower triangular integer matrices. For the ordinary Fibonacci case $(a=1=b)$ these positive integer triangles can be found in [5] under the nrs. A057995 and A057280. For the Pell case $(a=2, b=1)$ see nrs. A058402 and A058403.

Proposition 4 (Iteration of recurrence for $k-$fold convolutions; non-degenerate Lucas case):

For $a^2\, +\, 4\, b\, \neq\, 0$ the $k-$fold convolution of the generalized Lucas sequence $\{V_{n}(a,b)\}$ is expressed as linear combination of the two independent solutions of recurrence eq. 1 as given in eq. 40. The coefficient polynomials $AV_{k}(a,b;n)$ and $BV_{k}(a,b;n)$ satisfy the mixed recurrence relations eq. 41 and eq. 42.

Proof: Analogous to the one of Proposition 3. $\square$

Note 3: For integers $a$ and $b\neq\, -a^2/4$ the coefficients of the polynomials $AV_{n}(a,b;x)$ and $BV_{n}(a,b;x)$ furnish two lower triangular integer matrices. For the ordinary Lucas case $(a=1=b)$ these positive integer triangles can be found in [5] under the nrs. A061188 and A061189. For the Pell case $(a=2, b=1)$ see nrs. A062133 and A062134.