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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "B.C.Berndt: Number Theory in the Spirit of Ramanujanより\n",
    "\n",
    "Chapter 5 Corollary 5.1.7 \n",
    "\n",
    "$0 \\lt x \\lt 1$を満たす$x$について\n",
    "$$F\\left(\\frac{1}{2},\\frac{1}{2};1;1-\\frac{\\left(x-1\\right)^2}{\\left(x+1\\right)^2}\\right)=(1+x)\\,F\\left(\\frac{1}{2},\\frac{1}{2};1;x^2\\right)$$\n",
    "が成り立つ。\n",
    "\n",
    "\n",
    "\n",
    "証明の方針は単純です。$1-\\frac{\\left(x-1\\right)^2}{\\left(x+1\\right)^2}=\\frac{4\\,x}{\\left(x+1\\right)^2}$を示せば、\n",
    "$$F\\left(\\frac{1}{2},\\frac{1}{2};1;\\frac{4\\,x}{(1+x)^2}\\right)=(1+x)\\,F\\left(\\frac{1}{2},\\frac{1}{2};1;x^2\\right)$$\n",
    "より明らかです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}1-\\frac{\\left(x-1\\right)^2}{\\left(x+1\\right)^2}\\]"
      ],
      "text/plain": [
       "                                            2\n",
       "                                     (x - 1)\n",
       "(%o5)                            1 - --------\n",
       "                                            2\n",
       "                                     (x + 1)"
      ],
      "text/x-maxima": [
       "1-(x-1)^2/(x+1)^2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1-((x-1)/(x+1))^2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}\\frac{\\left(x+1\\right)^2-\\left(x-1\\right)^2}{\\left(x+1\\right)^2}\\]"
      ],
      "text/plain": [
       "                                     2          2\n",
       "                              (x + 1)  - (x - 1)\n",
       "(%o6)                         -------------------\n",
       "                                          2\n",
       "                                   (x + 1)"
      ],
      "text/x-maxima": [
       "((x+1)^2-(x-1)^2)/(x+1)^2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xthru(%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}\\frac{4\\,x}{\\left(x+1\\right)^2}\\]"
      ],
      "text/plain": [
       "                                     4 x\n",
       "(%o7)                              --------\n",
       "                                          2\n",
       "                                   (x + 1)"
      ],
      "text/x-maxima": [
       "(4*x)/(x+1)^2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(expand(%));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}F\\left( \\left. \\begin{array}{c}\\frac{1}{2},\\;\\frac{1}{2}\\\\1\\end{array} \\right |,1-\\frac{\\left(x-1\\right)^2}{\\left(x+1\\right)^2}\\right)=F\\left( \\left. \\begin{array}{c}\\frac{1}{2},\\;\\frac{1}{2}\\\\1\\end{array} \\right |,x^2\\right)\\,\\left(x+1\\right)\\]"
      ],
      "text/plain": [
       "                                             2\n",
       "                      1  1            (x - 1)\n",
       "(%o8) hypergeometric([-, -], [1], 1 - --------) = \n",
       "                      2  2                   2\n",
       "                                      (x + 1)\n",
       "                                                        1  1         2\n",
       "                                        hypergeometric([-, -], [1], x ) (x + 1)\n",
       "                                                        2  2"
      ],
      "text/x-maxima": [
       "hypergeometric([1/2,1/2],[1],1-(x-1)^2/(x+1)^2)\n",
       "  = hypergeometric([1/2,1/2],[1],x^2)*(x+1)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-HYPERGEOMETRIC in DEFUN\n"
     ]
    }
   ],
   "source": [
    "hypergeometric([1/2,1/2],[1],1-((x-1)/(x+1))^2)=(1+x)*hypergeometric([1/2,1/2],[1],x^2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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