{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "26100ad0",
   "metadata": {},
   "source": [
    "The investigation below suggests that the found polynomial families are appell sequences <=> are have translation properties <=> have interesting derivative properties \n",
    "\n",
    "1. https://en.wikipedia.org/wiki/Bernoulli_polynomials#Differences_and_derivatives\n",
    "2. https://en.wikipedia.org/wiki/Appell_sequence\n",
    "\n",
    "Q: are $p_m(n, k)$ linearly independent? Sol: show $p_m(0, k) = k^m$. Then this uses the fact the $x^m$ are linearly independent polynomials. Simple using the recurrence definition. \n",
    "\n",
    "Q: given a polynomial $p$ that is BH, is it a linear comb. of $\\{p_m\\}$? Given any polynomial in (n,k), is it a linear combination of $\\{p_m\\}$? Scratch: the bernoulli polynomials have an inversion formula, take a look at that. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ab663fcf",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "cc5e1c51",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')\n",
    "k = symbols('k')\n",
    "a = symbols('a')\n",
    "n = symbols('n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "id": "ff310a99",
   "metadata": {},
   "outputs": [],
   "source": [
    "import functools\n",
    "\n",
    "\n",
    "def bh(m):\n",
    "    if m < 0:\n",
    "        return lambda n, k: 0\n",
    "    if m == 0:\n",
    "        return lambda n, k: 1\n",
    "    elif m == 1:\n",
    "        return lambda n, k: k\n",
    "    else:\n",
    "        \n",
    "        def alternating_tangent(n):\n",
    "            # https://mathworld.wolfram.com/TangentNumber.html \n",
    "            return 2 ** (2*n) * (2**(2*n)-1) * Abs(bernoulli(2*n)) / (2*n)\n",
    "        \n",
    "        def R(n, k):\n",
    "            running_sum = 0\n",
    "            for j in range(1, round(m/2) + 1):\n",
    "                i = m - 2*j\n",
    "                running_sum += (-1)**(j) * alternating_tangent(j) * binomial(m-1, (2 * j - 1)) * bh(i)(n, k)\n",
    "            return running_sum\n",
    "        \n",
    "        def fm(n, k):\n",
    "            return k * bh(m-1)(n,k) + n * R(n, k)\n",
    "        \n",
    "        return fm\n",
    "    \n",
    "bh_ = lambda m: (lambda x: bh(m)(n, x)) # fixing n\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "id": "e41ddfff",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(m):\n",
    "    def fm(n, k):\n",
    "        return (k + n) ** (m)\n",
    "    return fm\n",
    "\n",
    "f_ = lambda m: (lambda x: f(m)(n, x)) # fixing n\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "71c8dce8",
   "metadata": {},
   "outputs": [],
   "source": [
    "for s in [f_, bh_]:\n",
    "    assert expand(s(2)(x-2)) == expand(s(2)(x) - 2 * s(1)(x) * 2 + s(0)(x) * 4)\n",
    "    assert expand(s(3)(x+1)) == expand(s(3)(x) + 3 * s(2)(x) + 3 * s(1)(x) + s(0)(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "id": "d4a6eae7",
   "metadata": {},
   "outputs": [],
   "source": [
    "# satifies appell seq. \n",
    "assert expand(diff(bh_(3)(x), x)) == expand(3 * bh_(2)(x))\n",
    "assert expand(diff(bh_(4)(x), x)) == expand(4 * bh_(3)(x))\n",
    "assert expand(diff(bh_(5)(x), x)) == expand(5 * bh_(4)(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "id": "d3400f51",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{4}$"
      ],
      "text/plain": [
       "k**4"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect( (expand(bh(4)(1, k+1) + bh(4)(1, k-1)))/2, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "id": "36385ebb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 k^{3} + 6 k^{2} + 12 k + 8$"
      ],
      "text/plain": [
       "2*k**3 + 6*k**2 + 12*k + 8"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(f(3)(1, k + 1) + f(3)(1, k - 1)) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "id": "833c7aa9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{3} - 3 k n + 3 k$"
      ],
      "text/plain": [
       "k**3 - 3*k*n + 3*k"
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(2 *binomial(3,0)*bh(3)(n, k) + 2 *binomial(3,1)*bh(1)(n, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "id": "4c4d41a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{4}$"
      ],
      "text/plain": [
       "k**4"
      ]
     },
     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(4)(0, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "id": "489fefe7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 k^{4} + 2 k^{3} - 12 k^{2} n - 6 k n + 6 n^{2} + 4 n$"
      ],
      "text/plain": [
       "2*k**4 + 2*k**3 - 12*k**2*n - 6*k*n + 6*n**2 + 4*n"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(4)(n+1, k+1) + bh(3)(n+1, k+1) +  bh(4)(n+1, k-1) + bh(3)(n+1, k-1) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "id": "070290f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 k^{4} + 2 k^{3} - 12 k^{2} n - 6 k n + 6 n^{2} + 4 n$"
      ],
      "text/plain": [
       "2*k**4 + 2*k**3 - 12*k**2*n - 6*k*n + 6*n**2 + 4*n"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(2 * bh(4)(n, k) + 2*bh(3)(n, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "id": "4bacbde4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{4} + 5 k^{3} - 6 k^{2} n + 3 k^{2} - 15 k n - 8 k + 3 n^{2} - n - 2$"
      ],
      "text/plain": [
       "k**4 + 5*k**3 - 6*k**2*n + 3*k**2 - 15*k*n - 8*k + 3*n**2 - n - 2"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(4)(n+1, k+1) + bh(3)(n+1, k+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "id": "01f7f470",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{4} - 3 k^{3} - 6 k^{2} n - 3 k^{2} + 9 k n + 8 k + 3 n^{2} + 5 n + 2$"
      ],
      "text/plain": [
       "k**4 - 3*k**3 - 6*k**2*n - 3*k**2 + 9*k*n + 8*k + 3*n**2 + 5*n + 2"
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(4)(n+1, k-1) + bh(3)(n+1, k-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "id": "160f9129",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{4} - 6 k^{2} n + 3 n^{2} + 2 n$"
      ],
      "text/plain": [
       "k**4 - 6*k**2*n + 3*n**2 + 2*n"
      ]
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(4)(n, k) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "id": "08d78c73",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{6} - 15 k^{4} n + 45 k^{2} n^{2} + 30 k^{2} n - 15 n^{3} - 30 n^{2} - 16 n$"
      ],
      "text/plain": [
       "k**6 - 15*k**4*n + 45*k**2*n**2 + 30*k**2*n - 15*n**3 - 30*n**2 - 16*n"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(6)(n, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "id": "e4158568",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1\n",
      "1 k\n",
      "2 k**2 - n\n",
      "3 k**3 - 3*k*n\n",
      "4 k**4 - 6*k**2*n + 3*n**2 + 2*n\n",
      "5 k**5 - 10*k**3*n + 15*k*n**2 + 10*k*n\n",
      "6 k**6 - 15*k**4*n + 45*k**2*n**2 + 30*k**2*n - 15*n**3 - 30*n**2 - 16*n\n"
     ]
    }
   ],
   "source": [
    "for m in range(7):\n",
    "    print(m, expand(bh(m)(n, k)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "id": "01df2d3a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle k^{6} - 15 k^{4} n + 45 k^{2} n^{2} + 30 k^{2} n - 15 n^{3} - 30 n^{2} - 16 n$"
      ],
      "text/plain": [
       "k**6 - 15*k**4*n + 45*k**2*n**2 + 30*k**2*n - 15*n**3 - 30*n**2 - 16*n"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(bh(6)(n, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "72ab3575",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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