Incomplete 𝑘-Fibonacci and 𝑘-Lucas Numbers

Table 1

The numbers 𝐹^𝑙_𝑘,𝑛, for 1⩽𝑛⩽10.

𝑛⧵𝑙	0	1	2	3	4
1	1
2	k
3	k ^2	𝑘^2+1
4	k ^3	𝑘^3+2𝑘
5	k ^4	𝑘^4+3𝑘^2	𝑘^4+3𝑘^2+1
6	k ^5	𝑘^5+4𝑘^3	𝑘^5+4𝑘^3+3𝑘
7	k ^6	𝑘^6+5𝑘^4	𝑘^6+5𝑘^4+6𝑘^2	𝑘^6+5𝑘^4+6𝑘^2+1
8	k ^7	𝑘^7+6𝑘^5	𝑘^7+6𝑘^5+10𝑘^3	𝑘^7+6𝑘^5+10𝑘^3+4𝑘
9	k ^8	𝑘^8+7𝑘^6	𝑘^8+7𝑘^6+15𝑘^4	𝑘^8+7𝑘^6+15𝑘^4+10𝑘^2	𝑘^8+7𝑘^6+15𝑘^4+10𝑘^2+1
10	k ^9	𝑘^9+8𝑘^7	𝑘^9+8𝑘^7+21𝑘^5	𝑘^9+8𝑘^7+21𝑘^5+20𝑘^3	𝑘^9+8𝑘^7+21𝑘^5+20𝑘^3+5𝑘