Example

(defun mulpoly (p1 p2)
  "p1, p2 - lists with coeff"
  (let* ((l1 (length p1))
    (l2 (length p2))
    (res (make-list (* l1 l2) :initial-element 0)))
    (loop for n1 below (length p1) do
      (loop for n2 below (length p2) do
        (incf (nth (+ n1 n2) res) (* (nth n1 p1) (nth n2 p2)))))
    res))

(defun product (x)
  (if (= x 0) 1
    (* x (product (1- x)))))

(defun c (a b)
  (/ (product a) (* (product b) (product (- a b)))))

(defun newton (a b n)
  "(a*x+b)^n to poly"
  (let ((res (make-list (1+ n) :initial-element 0)))
    (loop for k from 0 to n do
      (setf (nth k res)
        (* (expt a k) (c n k) (expt b (- n k)))))
    res))

(defun intpoly (p1)
  (let ((res (make-list (1+ (length p1)) :initial-element 0)))
    (loop for i below (length p1) do
      (setf (nth (1+ i) res) (/ (nth i p1) (length p1))))
    res))