Answered: Find a generating function for the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find a generating function for the number of integer solutions of 2x +3y+
7z =r with
(а) х, у, г > 0
(b) 0 < z <2< y < 8 < x

Expert Solution

Step 1

Here, we need to find a generating function for the number of integer solutions of $2 x + 3 y + 7 z = r$ with (a) $x, y, z \geq 0$ (b) $0 \leq z \leq 2 \leq y \leq 8 \leq x .$

We know that the generating function of $a x + b y + c z = d$ is

$(1 + x^{a} + {(x^{a})}^{2} + . . . + {(x^{a})}^{p} + . . . .) (1 + x^{b} + {(x^{b})}^{2} + . . . + {(x^{b})}^{q} + . . . .) (1 + x^{c} + {(x^{c})}^{2} + . . . + {(x^{c})}^{r} + . . . .)$

and the range of the series of p, q, and r will be depending on the condition.

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