const bessel = (x, alpha, n) => {
  if (n < 0) {
    throw new Error("n must be non-negative");
  }
  if (alpha < 0) {
    throw new Error("alpha must be non-negative");
  }

  let sum = 0;
  for (let m = 0; m <= n; m++) {
    const term =
      (Math.pow(-1, m) * Math.pow(x / 2, alpha + 2 * m)) /
      (factorial(m) * factorial(alpha + m));
    sum += term;
  }
  return sum;
};

const factorial = (n) => {
  if (n < 0) {
    throw new Error("n must be non-negative");
  }
  if (n === 0 || n === 1) {
    return 1;
  }
  let result = 1;
  for (let i = 2; i <= n; i++) {
    result *= i;
  }
  return result;
};

console.log(bessel(1, 0, 5));
console.log(bessel(1, 1, 5));
console.log(bessel(2, 0, 5));