q-analog.ipynb

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    "<h3>q-analog</h3>\n",
    "<h4>q-Pochhammer symbol</h4>\n",
    "<h5>mpmath.qp</h5>\n",
    "<a href=\"https://en.wikipedia.org/wiki/Q-Pochhammer_symbol\">q-Pochhammer symbol</a>\n",
    "$$(a;q)_n=\\prod_{k=0}^{n-1}1-aq^k$$\n",
    "<p>where</p>\n",
    "$$(a;q)_0=1$$\n",
    "<h5>negative n</h5>\n",
    "$$(a;q)_{-n}=\\frac1{(aq^{-n};q)_n}$$\n",
    "$$=\\prod_{k=1}^n\\frac1{1-\\frac{a}{q^k}}$$\n",
    "<h5>n=&infin;</h5>\n",
    "<a href=\"https://en.wikipedia.org/wiki/Q-Pochhammer_symbol#Identities\">infinite product</a>\n",
    "$$(x;q)_\\infty=\\sum_{n=0}^\\infty\\frac{-1^n q^{n(n-1)/2}}{(q;q)_n}x^n$$\n",
    "<p>Gasper & Rahman[1990]</p>\n",
    "<p>when |q|&lt;1</p>\n",
    "<p>if |q|&gt;1 an inversion with respect to q can be performed by setting p=q<sup>-1</sup><p>\n",
    "$$(a;q)_n=(a^{-1};p)_n(-a)^n p^{-{n\\choose 2}}\\quad(1.2.24)$$\n",
    "\n",
    "<a href=\"https://en.wikipedia.org/wiki/Q-Pochhammer_symbol#Relationship_to_other_q-functions\">q-analog</a>\n",
    "<h4>q-bracket</h4>\n",
    "$$[n]_q=\\frac{1-q^n}{1-q}$$\n",
    "<a href=\"https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient#Definition\">q-bracket</a>\n",
    "$$[k]_q=\\sum_{0\\leq i<k}q^i=\n",
    "\\begin{cases}\n",
    "\\frac{1-q^k}{1-q} & \\text{for} & q \\neq 1 \\\\\n",
    "k & \\text{for} & q = 1 \n",
    "\\end{cases}$$\n",
    "<h4>q-factorial</h4>\n",
    "<h5>mpmath.qfac</h5>\n",
    "$$[n]!_q=\\prod_{k=1}^n [k]_q\\quad(1)$$\n",
    "<br>\n",
    "$$=\\frac{(q;q)_n}{(1-q)^n}\\quad(2)$$\n",
    "<h4>q-binomial coefficients</h4>\n",
    "$$\\begin{bmatrix}\n",
    "n\\\\\n",
    "k\n",
    "\\end{bmatrix}_q\\text{ or }{n\\choose k}_q$$\n",
    "<br>\n",
    "$$\\begin{bmatrix}\n",
    "n\\\\\n",
    "k\n",
    "\\end{bmatrix}_q\n",
    "=\\frac{[n]!_q}{[n-k]!_q [k]!_q}$$\n",
    "<h4>basic hypergeometric series</h4>\n",
    "<a href=\"https://en.wikipedia.org/wiki/Basic_hypergeometric_series#Definition\">basic hypergeometric series</a>\n",
    "<h5>unilateral basic hypergeometric series φ</h5>\n",
    "<h5>mpmath.qhyper</h5>\n",
    "$$_{j}\\phi_k \\left[\\begin{matrix} \n",
    "a_1 & a_2 & \\ldots & a_{j}\\\\ \n",
    "b_1 & b_2 & \\ldots & b_k \\end{matrix} \n",
    ";q,z\\right]=\\sum_{n=0}^\\infty  \n",
    "\\frac{(a_1,a_2,\\ldots,a_{j};q)_n}{(b_1,b_2,\\ldots,b_k,q;q)_n}\\left(-1^nq^{n\\choose 2}\\right)^{1+k-j}z^n$$\n",
    "<p>where</p>\n",
    "$$(a_1,a_2,\\ldots,a_m;q)_n=(a_1;q)_n(a_2;q)_n\\ldots(a_m;q)_n$$\n",
    "<a href=\"https://mathworld.wolfram.com/q-HypergeometricFunction.html\">q-hypergeometric function</a>\n",
    "<p><a href=\"https://www.researchgate.net/publication/225384247_The_Andrews-Gordon_Identities_and_q-Multinomial_Coefficients\">Warnaar, 1997</a></p>\n",
    "$$\\text{binomial coefficient }{n\\choose 2}=\\tfrac12n(n-1)$$\n",
    "$${n\\choose k}={n\\choose k}_1\\quad(2.2)$$\n",
    "<h5>bilateral basic hypergeometric series ψ</h5>\n",
    "$$_j\\psi_k \\left[\\begin{matrix} \n",
    "a_1 & a_2 & \\ldots & a_j\\\\ \n",
    "b_1 & b_2 & \\ldots & b_k\\end{matrix} \n",
    ";q,z\\right]=\\sum_{n=-\\infty}^\\infty  \n",
    "\\frac{(a_1,a_2,\\ldots,a_j;q)_n}{(b_1,b_2,\\ldots,b_k;q)_n}\\left(-1^nq^{n\\choose 2}\\right)^{k-j}z^n$$\n",
    "<a href=\"https://dlmf.nist.gov/17.4#E3\">bilateral basic hypergeometric series</a>\n",
    "$$=\\sum_{n=0}^\\infty  \n",
    "\\frac{(a_1,a_2,\\ldots,a_j;q)_n}{(b_1,b_2,\\ldots,b_k;q)_n}\\left(-1^nq^{n\\choose 2}\\right)^{k-j}z^n+\\sum_{n=1}^\\infty  \n",
    "\\frac{(q/b_1,q/b_2,\\ldots,q/b_k;q)_n}{(q/a_1,q/a_2,\\ldots,q/a_j;q)_n}\\left(\\frac{b_1b_2\\ldots b_k}{a_1a_2\\ldots a_jz}\\right)^n\\quad(17.4.3)$$\n",
    "<h4>q-gamma function</h4>\n",
    "<a href=\"https://dlmf.nist.gov/5.18#ii\">q-gamma function</a>\n",
    "$$\\Gamma_q(z)=\\frac{(q;q)_\\infty(1-q)^{1-z}}{(q^z;q)_\\infty}\\quad(5.18.4)$$\n",
    "<p>when 0&lt;q&lt;1</p>\n",
    "$$\\Gamma_q(1)=1\\quad(5.18.5)$$\n",
    "$$\\Gamma_q(2)=1\\quad(5.18.5)$$\n",
    "<a href=\"https://en.wikipedia.org/wiki/Q-gamma_function\">q-gamma function</a>\n",
    "$$\\Gamma_q(x)=\\frac{(q^{-1};q^{-1})_\\infty}{(q^{-x};q^{-1})_\\infty}(q-1)^{1-x}q^{\\binom{x}{2}}$$\n",
    "<p>when q&gt;1</p>\n",
    "<h4>Jackson q-Bessel function</h4>\n",
    "<a href=\"https://en.wikipedia.org/wiki/Jackson_q-Bessel_function#Definition\">Jackson q-Bessel function</a>\n",
    "$$J_\\nu^{(1)}(x;q)=\\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{x}2\\right)^\\nu{}_2\\phi_1\\left(0,0;q^{\\nu+1};q,\\tfrac{-x^2}4\\right),\\quad |x|<2$$\n",
    "$$J_\\nu^{(2)}(z;q)=\\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{z}2\\right)^\\nu{}_0\\phi_1\\left(;q^{\\nu+1};q,\\tfrac{-z^2q^{\\nu +1}}4\\right)$$\n",
    "$$J_\\nu^{(3)}(z;q)=\\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{z}2\\right)^\\nu{}_1\\phi_1\\left(0;q^{\\nu+1};q,\\tfrac{qz^2}4\\right)$$\n",
    "<h5>negative argument of integer order</h5>\n",
    "$$J_\\nu^{(1)}(-x;q)=-1^\\nu J_\\nu^{(1)}(x;q)$$\n",
    "$$J_\\nu^{(2)}(-z;q)=-1^\\nu J_\\nu^{(2)}(z;q)$$\n",
    "$$J_\\nu^{(3)}(-z;q)=-1^\\nu J_\\nu^{(3)}(z;q)$$\n",
    "<h5>negative integer order</h5>\n",
    "$$J_{-\\nu}^{(1)}(x;q)=-1^\\nu J_\\nu^{(1)}(x;q)$$\n",
    "$$J_{-\\nu}^{(2)}(z;q)=-1^\\nu J_\\nu^{(2)}(z;q)$$\n",
    "<h5>modified q-Bessel functions</h5>\n",
    "<a href=\"https://en.wikipedia.org/wiki/Jackson_q-Bessel_function#Modified_q-Bessel_Functions\">modified q-Bessel functions</a>\n",
    "$$I_\\nu^{(1)}(x;q)=e^{i\\nu\\pi/2}J_{\\nu}^{(1)}(x;q)$$\n",
    "$$I_\\nu^{(2)}(x;q)=e^{i\\nu\\pi/2}J_{\\nu}^{(2)}(x;q)$$\n",
    "<br>\n",
    "$$K_\\nu^{(1)}(x;q)=\\frac{\\pi}{2\\sin(\\pi\\nu)}\\left\\{I_{-\\nu}^{(1)}(x;q)-I_\\nu^{(1)}(x;q)\\right\\}$$\n",
    "<br>\n",
    "$$K_\\nu^{(2)}(x;q)=\\frac{\\pi}{2\\sin(\\pi\\nu)}\\left\\{I_{-\\nu}^{(2)}(x;q)-I_\\nu^{(2)}(x;q)\\right\\}$$\n",
    "<a href=\"https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-121\">Mahmoud, 2013</a>\n",
    "<p>unnumbered equation by Rogov, after (9)</p>\n",
    "$$I_n^{(1)}(x;q)=\\frac{(q^{n+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{x}2\\right)^n{}_2\\phi_1\\left(0,0;q^{n+1};q,\\tfrac{x^2}4\\right)$$\n",
    "$$I_n^{(2)}(x;q)=\\frac{(q^{n+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{x}2\\right)^n{}_0\\phi_1\\left(-;q^{n+1};q,\\tfrac{q^{2(n+1)/2}x^2}{4}\\right)$$\n",
    "$$I_n^{(3)}(x;q)=\\frac{(q^{n+1};q)_\\infty}{(q;q)_\\infty}\\left(\\tfrac{x}2\\right)^n{}_1\\phi_1\\left(0;q^{n+1};q,\\tfrac{q^{(n+1)/2}x^2}{4}\\right)$$\n",
    "<h5>Jackson q-difference operator D<sub>q</sub></h5>\n",
    "<a href=\"https://advancesindifferenceequations.springeropen.com/articles/10.1186/1687-1847-2013-121\">Mahmoud, 2013</a>\n",
    "$$D_qf(x)=\\frac{f(x)-f(qx)}{(1-q)x}\\quad(3)$$\n",
    "<br>\n",
    "$$D_qJ_\\nu^{(1)}(x;q)=\\frac{J_\\nu^{(1)}(x;q)-J_\\nu^{(1)}(qx;q)}{(1-q)x}$$\n",
    "$$D_qJ_\\nu^{(2)}(x;q)=\\frac{J_\\nu^{(2)}(x;q)-J_\\nu^{(2)}(qx;q)}{(1-q)x}$$\n",
    "$$D_qJ_\\nu^{(3)}(x;q)=\\frac{J_\\nu^{(3)}(x;q)-J_\\nu^{(3)}(qx;q)}{(1-q)x}$$\n",
    "$$D_qI_\\nu^{(1)}(x;q)=\\frac{I_\\nu^{(1)}(x;q)-I_\\nu^{(1)}(qx;q)}{(1-q)x}$$\n",
    "$$D_qI_\\nu^{(2)}(x;q)=\\frac{I_\\nu^{(2)}(x;q)-I_\\nu^{(3)}(qx;q)}{(1-q)x}$$\n",
    "$$D_qI_\\nu^{(3)}(x;q)=\\frac{I_\\nu^{(3)}(x;q)-I_\\nu^{(3)}(qx;q)}{(1-q)x}$$\n",
    "<h5>symmetric q-difference operator &delta;<sub>q</sub></h5>\n",
    "<p>Gasper and Rahman, \"Basic Hypergeometric Series\", 2nd ed., 1990.</p>\n",
    "$$\\delta_qf(z)=f\\left(z\\sqrt{q}\\right)-f\\left(\\frac{z}{\\sqrt{q}}\\right)\\quad(1.3.23)$$\n",
    "<h5>forward q-difference operator &Delta;</h5>\n",
    "$$\\Delta f(z)=f(qz)-f(z)\\quad(1.3.20)$$\n",
    "<h5>backward q-difference operator ∇</h5>\n",
    "$$\\nabla f(z)=f\\left(q^{-1}z\\right)-f(z)\\quad(1.3.20)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [],
   "source": [
    "from typing import List, TypeVar\n",
    "\n",
    "Numeric = TypeVar('Numeric', int, float, complex)\n",
    "\n",
    "EMPTY_PRODUCT = 1\n",
    "EMPTY_SUM = 0\n",
    "MINUS_ONE = -1\n",
    "\n",
    "def q_poch(a: Numeric, q: Numeric, n: int=None) -> Numeric:\n",
    "    \"\"\"Pure Python q-Pochhammer symbol (a;q)_n is the q-analog of Pochhammer symbol.\n",
    "    Also called q-shifted factorial.\n",
    "    (a;q)_n = q_poch(a, q, n)\n",
    "    (a;q)_infinity = q_poch(a, q)\n",
    "    \"\"\"\n",
    "    #Special case of q-binomial thereom.\n",
    "    if n is None:\n",
    "        sum = EMPTY_SUM\n",
    "        for n in range(30):\n",
    "            sum += MINUS_ONE**n*q**(n*(n - 1)/2)/q_poch(q, q, n)*a**n\n",
    "        return sum\n",
    "    if not n:\n",
    "        return 1\n",
    "    signum_n = 1\n",
    "    if n < 0:\n",
    "        n = abs(n)\n",
    "        signum_n = -1\n",
    "    product = EMPTY_PRODUCT\n",
    "    if signum_n == 1:\n",
    "        for k in range(n):\n",
    "            product *= 1 - a*q**k\n",
    "    else:\n",
    "        for k in range(1, n + 1):\n",
    "            product *= 1/(1 - a/q**k)\n",
    "    return product\n",
    "\n",
    "\n",
    "def q_bracket(n: int, q: Numeric) -> Numeric:\n",
    "    \"\"\"Pure Python q_bracket of n [n]_q is the q-analog of n.\n",
    "    Also called q-number of n.\n",
    "    \"\"\"\n",
    "    if q == 1:\n",
    "        return n\n",
    "    return (1 - q**n)/(1 - q)\n",
    "\n",
    "\n",
    "def q_factorial(n: int, q: Numeric, type: int=1) -> Numeric:\n",
    "    \"\"\"Pure Python q_factorial [n]!_q is the q-analog of factorial.\n",
    "    type=1 is algorithm (1).\n",
    "    type=2 is algorithm (2).\n",
    "    \"\"\"\n",
    "    product = EMPTY_PRODUCT\n",
    "    if type == 1:\n",
    "        for k in range(1, n + 1):\n",
    "            product *= q_bracket(k, q)\n",
    "        return product\n",
    "    elif type == 2:\n",
    "        return q_poch(q, q, n)/(1 - q)**n\n",
    "\n",
    "\n",
    "def q_binom(n: int, k: int, q: Numeric) -> Numeric:\n",
    "    \"\"\"Pure Python q-binomial coefficients [n choose k]_q is the q-analog of (n choose k).\n",
    "    Also called Gaussian binomial coefficients, Gaussian coefficients or Gaussian polynomials.\n",
    "    \"\"\"\n",
    "    return q_factorial(n, q)/(q_factorial(n - k, q)*q_factorial(k, q))\n",
    "\n",
    "\n",
    "def binom(n: int, k: int) -> int:\n",
    "    \"\"\"Pure Python binomial coefficient (n choose k) using q_binom function.\"\"\"\n",
    "    return int(q_binom(n, k, 1))\n",
    "\n",
    "\n",
    "def q_hyper(a: List[Numeric], b: List[Numeric], q: Numeric, z: Numeric, type: int=1) -> Numeric:\n",
    "    \"\"\"Pure Python unilateral basic hypergeometric series φ is the q-analog\n",
    "    of generalized hypergeometric function.\n",
    "    Also called q-hypergeometric series or q-hypergeometric function.\n",
    "    type=1 is unilateral basic hypergeometric series φ.\n",
    "    type=2 is bilateral basic hypergeometric series ψ.\n",
    "    \"\"\"\n",
    "    sum = EMPTY_SUM\n",
    "    j = len(a)\n",
    "    k = len(b)\n",
    "    if type == 1:\n",
    "        for n in range(30):\n",
    "            a_product = EMPTY_PRODUCT\n",
    "            b_product = EMPTY_PRODUCT\n",
    "            for a_item in a:\n",
    "                a_product *= q_poch(a_item, q, n)\n",
    "            for b_item in b:\n",
    "                b_product *= q_poch(b_item, q, n)\n",
    "            b_product *= q_poch(q, q, n)\n",
    "            sum += a_product/b_product*(MINUS_ONE**n*q**binom(n, 2))**(1 + k - j)*z**n\n",
    "    if type == 2:\n",
    "        sum_1 = EMPTY_SUM\n",
    "        sum_2 = EMPTY_SUM\n",
    "        for n in range(30):\n",
    "            a_product = EMPTY_PRODUCT\n",
    "            b_product = EMPTY_PRODUCT\n",
    "            aq_product = EMPTY_PRODUCT\n",
    "            bq_product = EMPTY_PRODUCT\n",
    "            for a_item in a:\n",
    "                a_product *= q_poch(a_item, q, n)\n",
    "                #skip n=0 for sum_2, i.e. sum_2 starts at n=1\n",
    "                if n:\n",
    "                    aq_product *= q/q_poch(a_item, q, n)\n",
    "            for b_item in b:\n",
    "                b_product *= q_poch(b_item, q, n)\n",
    "                #skip n=0 for sum_2, i.e. sum_2 starts at n=1\n",
    "                if n:\n",
    "                    bq_product *= q/q_poch(b_item, q, n)\n",
    "            sum_1 += a_product/b_product*(MINUS_ONE**n*q**binom(n, 2))**(k - j)*z**n\n",
    "            sum_2 += bq_product/aq_product*(b_item/(a_item*z))**n\n",
    "        sum = sum_1 + sum_2\n",
    "    return sum\n",
    "\n",
    "\n",
    "def q_gamma(z: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-gamma function Γ_q is the q-analog of gamma function.\n",
    "    Using q inversion, supports q>1.\n",
    "    \"\"\"\n",
    "    if q > 1:\n",
    "        #q inversion\n",
    "        return q_poch(q**-1, q**-1)/q_poch(q**-z, q**-1)*(1 - q)**(1 - z)*binom(z, 2)\n",
    "    return q_poch(q, q)*(1 - q)**(1 - z)/q_poch(q**z, q)\n",
    "\n",
    "\n",
    "def q_besselj1(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"first Jackson q-Bessel function J^(1)_𝜈 is one of three q-analogs for Bessel function\n",
    "    of the first kind.\n",
    "    Also called first basic Bessel function.\n",
    "    \"\"\"\n",
    "    z_signum = 1\n",
    "    if z.real < 0:\n",
    "        if not z.imag:\n",
    "            z = -z\n",
    "            z_signum = -1\n",
    "    𝜈_signum = 1\n",
    "    if 𝜈 == int(𝜈.real):\n",
    "        if not 𝜈.imag:\n",
    "            if 𝜈 < 0:\n",
    "                𝜈 = -𝜈\n",
    "                𝜈_signum = -1\n",
    "    result = q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([0, 0], [q**(𝜈 + 1)], q, -z**2/4)\n",
    "    if 𝜈_signum == -1:\n",
    "        return result*z_signum**𝜈*𝜈_signum**𝜈\n",
    "    return result*z_signum**𝜈\n",
    "\n",
    "def q_besselj2(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"second Jackson q-Bessel function J^(2)_𝜈 is one of three q-analogs for Bessel function\n",
    "    of the first kind.\n",
    "    Also called second basic Bessel function.\n",
    "    \"\"\"\n",
    "    z_signum = 1\n",
    "    if z.real < 0:\n",
    "        if not z.imag:\n",
    "            z = -z\n",
    "            z_signum = -1\n",
    "    𝜈_signum = 1\n",
    "    if 𝜈 == int(𝜈.real):\n",
    "        if not 𝜈.imag:\n",
    "            if 𝜈 < 0:\n",
    "                𝜈 = -𝜈\n",
    "                𝜈_signum = -1\n",
    "    result = q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([], [q**(𝜈 + 1)], q, (-z**2*q**(𝜈 + 1))/4)\n",
    "    if 𝜈_signum == -1:\n",
    "        return result*z_signum**𝜈*𝜈_signum**𝜈\n",
    "    return result*z_signum**𝜈\n",
    "\n",
    "def q_besselj3(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"third Jackson q-Bessel function J^(3)_𝜈 is one of three q-analogs for Bessel function\n",
    "    of the first kind.\n",
    "    Also called Hahn–Exton q-Bessel function or third basic Bessel function.\n",
    "    \"\"\"\n",
    "    z_signum = 1\n",
    "    if z.real < 0:\n",
    "        if not z.imag:\n",
    "            z = -z\n",
    "            z_signum = -1\n",
    "    result = q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([0], [q**(𝜈 + 1)], q, (q*z**2)/4)\n",
    "    return result*z_signum**𝜈\n",
    "\n",
    "\n",
    "def q_besseli1(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"first modified Jackson q-Bessel function I^(1)_𝜈 is one of three q-analogs for modified Bessel functionn\n",
    "    of the first kind.\n",
    "    Also called first modified basic Bessel function and unified as one equation and called\n",
    "    generalized modified q-Bessel function.\n",
    "    \"\"\"\n",
    "    return q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([0, 0], [q**(𝜈 + 1)], q, z**2/4)\n",
    "\n",
    "\n",
    "def q_besseli2(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"second modified Jackson q-Bessel function I^(2)_𝜈 is one of three q-analogs for modified Bessel functionn\n",
    "    of the first kind.\n",
    "    Also called second modified basic Bessel function and unified as one equation and called\n",
    "    generalized modified q-Bessel function.\n",
    "    \"\"\"\n",
    "    return q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([], [q**(𝜈 + 1)], q, q**(2*(𝜈 + 1)/2)*z**2/4)\n",
    "\n",
    "\n",
    "def q_besseli3(z: Numeric, q: Numeric, 𝜈: int) -> Numeric:\n",
    "    \"\"\"third modified Jackson q-Bessel function I^(3)_𝜈 is one of three q-analogs for modified Bessel functionn\n",
    "    of the first kind.\n",
    "    Also called third modified basic Bessel function and unified as one equation and called\n",
    "    generalized modified q-Bessel function.\n",
    "    \"\"\"\n",
    "    return q_poch(q**(𝜈 + 1), q)/q_poch(q, q)*(z/2)**𝜈*q_hyper([0], [q**(𝜈 + 1)], q, q**((𝜈 + 1)/2)*z**2/4)\n",
    "\n",
    "\n",
    "def q_besselj1p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"q-difference operator applied to first Jackson q-Bessel function J^(1)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besselj1p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besselj1p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besselj1(z, q, 𝜈) - q_besselj1(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besselj1(z*q**0.5, q, 𝜈) - q_besselj1(z/q**0.5, q, 𝜈)\n",
    "\n",
    "\n",
    "def q_besselj2p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"q-difference operator applied to second Jackson q-Bessel function J^(2)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besselj2p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besselj2p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besselj2(z, q, 𝜈) - q_besselj2(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besselj2(z*q**0.5, q, 𝜈) - q_besselj2(z/q**0.5, q, 𝜈)\n",
    "\n",
    "\n",
    "def q_besselj3p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"Jackson q-difference operator applied to third Jackson q-Bessel function J^(3)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besselj3p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besselj3p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besselj3(z, q, 𝜈) - q_besselj3(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besselj3(z*q**0.5, q, 𝜈) - q_besselj3(z/q**0.5, q, 𝜈)\n",
    "\n",
    "\n",
    "def q_besseli1p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"Jackson q-difference operator applied to first Jackson q-Bessel function I^(1)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besseli1p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besseli1p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besseli1(z, q, 𝜈) - q_besseli1(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besseli1(z*q**0.5, q, 𝜈) - q_besseli1(z/q**0.5, q, 𝜈)\n",
    "\n",
    "\n",
    "def q_besseli2p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"Jackson q-difference operator applied to second Jackson q-Bessel function I^(2)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besseli2p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besseli2p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besseli2(z, q, 𝜈) - q_besseli2(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besseli2(z*q**0.5, q, 𝜈) - q_besseli2(z/q**0.5, q, 𝜈)\n",
    "\n",
    "\n",
    "def q_besseli3p(z: Numeric, q: Numeric, 𝜈: int, type: int=1) -> Numeric:\n",
    "    \"\"\"Jackson q-difference operator applied to third Jackson q-Bessel function I^(3)_𝜈 is one of three q-analogs for first derivative of Bessel function\n",
    "    of the first kind.\n",
    "    q_besseli3p(type=1) Jackson q-difference operator D_q.\n",
    "    q_besseli3p(type=2) symmetric q-difference operator 𝛿_q.\n",
    "    \"\"\"\n",
    "    if type == 1:\n",
    "        return (q_besseli3(z, q, 𝜈) - q_besseli3(q*z, q, 𝜈))/((1 - q)*z)\n",
    "    elif type == 2:\n",
    "        return q_besseli3(z*q**0.5, q, 𝜈) - q_besseli3(z/q**0.5, q, 𝜈)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.891931730935808\n",
      "1\n",
      "1.0833687838945465e-11\n",
      "9.889595851767801\n",
      "2.893319969218555\n",
      "62.267381048088886\n",
      "(0.46288422316601435+4.481821753552705j)\n",
      "1.0\n",
      "1.2\n",
      "7657.445986770649\n",
      "7657.44598677065\n",
      "0.004664357446868912\n",
      "0.004664357446868914\n",
      "1.0 1\n",
      "1.2 1.2\n",
      "1.24 1.24\n",
      "10\n",
      "-0.19758490912633556\n",
      "(2.8063302449257166+3.5689976233379435j)\n",
      "(9.112885171773401-1.272756997166375j)\n",
      "-3.5967458976487066e+17\n",
      "(-8.84046098239473e+16+1.4359750151402227e+17j)\n",
      "(13.785428321405657-0.6371154937666327j)\n",
      "3.868857608488975e+16\n",
      "-32653207226340.957\n",
      "1.0 1\n",
      "1.0000000000000002 1\n",
      "1.1999999999999997\n",
      "(0.8148277076382674+0.3936306471345367j)\n",
      "0.144\n",
      "0.908570685083017\n",
      "0.2886386045267251\n",
      "0.0753642470125165\n",
      "(-571093958024617.6-276654478837738.97j)\n",
      "0.9912224383093116\n",
      "-0.9314911465917114\n",
      "290.3577345283539\n",
      "0.908570685083017\n",
      "-0.2886386045267251\n",
      "0.0753642470125165\n",
      "0.9912224383093116\n",
      "0.9314911465917114\n",
      "290.3577345283539\n",
      "-0.2886386045267251\n",
      "0.0753642470125165\n",
      "0.9912224383093116\n",
      "-0.9314911465917114\n",
      "290.3577345283539\n",
      "0.980479346165934\n",
      "0.3114828542137617\n",
      "0.08132893659051789\n",
      "(-1.228958577417209+0.6633523714326358j)\n",
      "(-569343586928796.1-294947966318273.56j)\n",
      "1.0243570468451848\n",
      "-0.9626290560094936\n",
      "300.06382016274483\n",
      "0.980479346165934\n",
      "-0.3114828542137617\n",
      "0.08132893659051789\n",
      "1.0243570468451848\n",
      "0.9626290560094936\n",
      "300.06382016274483\n",
      "-0.3114828542137617\n",
      "0.08132893659051789\n",
      "0.980521704989218\n",
      "0.30742708253404594\n",
      "0.08010172975324056\n",
      "(-1.428677603620305+0.9751516220061113j)\n",
      "(-574656250939058.4-273612157188393.38j)\n",
      "1.0081995665422878\n",
      "-0.7997321411255556\n",
      "295.96451186220156\n",
      "0.980521704989218\n",
      "-0.30742708253404594\n",
      "0.08010172975324056\n",
      "1.1047277452232445\n",
      "0.3397127118615385\n",
      "0.08823533633160541\n",
      "(-4.601792676611349-3.8808962869376638j)\n",
      "(-574958341297561.6-278680105994935.16j)\n",
      "1.019541846596653\n",
      "0.31351735942295783\n",
      "0.08143148221593967\n",
      "(-1.2124487317792683+0.6387414211984294j)\n",
      "(-575806402021513.8-260140530977591.06j)\n",
      "0.9565914059401194\n",
      "0.30742708253404594\n",
      "0.08080762939412182\n",
      "(-1.3133221832302016+0.7919012488748802j)\n",
      "(-578596504233014.5-274935323923937j)\n",
      "-0.21883408611418548\n",
      "0.5658538057513636\n",
      "0.18029816640400065\n",
      "-0.04684855197065574\n",
      "0.6224774805010329\n",
      "0.19518452573404102\n",
      "-0.046742824450274056\n",
      "0.6124194209139112\n",
      "0.19212143029610212\n",
      "0.2520271796357909\n",
      "0.6925218147434156\n",
      "0.2124246165518643\n",
      "0.04690144910607319\n",
      "0.6275230534198565\n",
      "0.1954404796150938\n",
      "-0.10415522447112135\n",
      "0.6124194209139112\n",
      "0.193883345534875\n",
      "0.3434647865374909\n",
      "-0.34781629322959745\n",
      "-0.270355005404136\n",
      "0.09348554124103547\n",
      "-0.5477407196442877\n",
      "-0.3893453643225885\n",
      "0.09243093470519681\n",
      "-0.5033500682315808\n",
      "-0.3590476129716727\n",
      "-0.7106090482892353\n",
      "-0.977192833806403\n",
      "-0.6267921457590694\n",
      "-0.09401451259541815\n",
      "-0.570305207939647\n",
      "-0.39190490313415116\n",
      "0.2030568081411478\n",
      "-0.5033500682315808\n",
      "-0.3764002290888586\n"
     ]
    }
   ],
   "source": [
    "print(q_poch(-1.2, 0.2, 5))\n",
    "print(q_poch(-1.2, 0.2, 0))\n",
    "print(q_poch(-1.2, 0.2, -5))\n",
    "print(q_poch(-1.2, -1.2, 5))\n",
    "print(q_poch(-1.2, 0.2))\n",
    "print(q_poch(-1.2, -1.2))\n",
    "print(q_poch(2-1j, 0+0.75j))\n",
    "print(q_bracket(1, 0.2))\n",
    "print(q_bracket(2, 0.2))\n",
    "print(q_factorial(10, 0.7))\n",
    "print(q_factorial(10, 0.7, type=2))\n",
    "print(q_factorial(10, -0.7))\n",
    "print(q_factorial(10, -0.7, type=2))\n",
    "print(q_binom(1, 1, 0.2), 1)\n",
    "print(q_binom(2, 1, 0.2), 1 + 0.2)\n",
    "print(q_binom(3, 1, 0.2), 1 + 0.2 + 0.2**2)\n",
    "print(binom(5, 2))\n",
    "print(q_hyper([0.5], [2.25], 0.25, 4))\n",
    "print(q_hyper([0.5], [2.25], 0.25-0.25j, 4))\n",
    "print(q_hyper([1+1j], [2,3+0.5j], 0.25, 3+4j))\n",
    "print(q_hyper([0.5], [2.25], 0.25, 4, type=2))\n",
    "print(q_hyper([0.5], [2.25], 0.25-0.25j, 4, type=2))\n",
    "print(q_hyper([1+1j], [2,3+0.5j], 0.25, 3+4j, type=2))\n",
    "print(q_gamma(0, 0.2))\n",
    "print(q_gamma(-2, 0.2))\n",
    "print(q_gamma(1, 0.2), 1)\n",
    "print(q_gamma(2, 0.2), 1)\n",
    "print(q_gamma(3, 0.2))\n",
    "print(q_gamma(2+2j, 0.2))\n",
    "print(q_gamma(3, 5))\n",
    "print(q_besselj1(0.5, 0.2, 0))\n",
    "print(q_besselj1(0.5, 0.2, 1))\n",
    "print(q_besselj1(0.5, 0.2, 2))\n",
    "print(q_besselj1(0.5, 0.2+2j, 2))\n",
    "print(q_besselj1(0.5, -1.2, 0))\n",
    "print(q_besselj1(0.5, -1.2, 1))\n",
    "print(q_besselj1(0.5, -1.2, 2))\n",
    "print(q_besselj1(-0.5, 0.2, 0))\n",
    "print(q_besselj1(-0.5, 0.2, 1))\n",
    "print(q_besselj1(-0.5, 0.2, 2))\n",
    "print(q_besselj1(-0.5, -1.2, 0))\n",
    "print(q_besselj1(-0.5, -1.2, 1))\n",
    "print(q_besselj1(-0.5, -1.2, 2))\n",
    "print(q_besselj1(0.5, 0.2, -1))\n",
    "print(q_besselj1(0.5, 0.2, -2))\n",
    "print(q_besselj1(0.5, -1.2, 0))\n",
    "print(q_besselj1(0.5, -1.2, 1))\n",
    "print(q_besselj1(0.5, -1.2, 2))\n",
    "print(q_besselj2(0.5, 0.2, 0))\n",
    "print(q_besselj2(0.5, 0.2, 1))\n",
    "print(q_besselj2(0.5, 0.2, 2))\n",
    "print(q_besselj2(0.5+2j, 0.2, 2))\n",
    "print(q_besselj2(0.5, 0.2+2j, 2))\n",
    "print(q_besselj2(0.5, -1.2, 0))\n",
    "print(q_besselj2(0.5, -1.2, 1))\n",
    "print(q_besselj2(0.5, -1.2, 2))\n",
    "print(q_besselj2(-0.5, 0.2, 0))\n",
    "print(q_besselj2(-0.5, 0.2, 1))\n",
    "print(q_besselj2(-0.5, 0.2, 2))\n",
    "print(q_besselj2(-0.5, -1.2, 0))\n",
    "print(q_besselj2(-0.5, -1.2, 1))\n",
    "print(q_besselj2(-0.5, -1.2, 2))\n",
    "print(q_besselj2(0.5, 0.2, -1))\n",
    "print(q_besselj2(0.5, 0.2, -2))\n",
    "print(q_besselj3(0.5, 0.2, 0))\n",
    "print(q_besselj3(0.5, 0.2, 1))\n",
    "print(q_besselj3(0.5, 0.2, 2))\n",
    "print(q_besselj3(0.5+2j, 0.2, 2))\n",
    "print(q_besselj3(0.5, 0.2+2j, 2))\n",
    "print(q_besselj3(0.5, -1.2, 0))\n",
    "print(q_besselj3(0.5, -1.2, 1))\n",
    "print(q_besselj3(0.5, -1.2, 2))\n",
    "print(q_besselj3(-0.5, 0.2, 0))\n",
    "print(q_besselj3(-0.5, 0.2, 1))\n",
    "print(q_besselj3(-0.5, 0.2, 2))\n",
    "print(q_besseli1(0.5, 0.2, 0))\n",
    "print(q_besseli1(0.5, 0.2, 1))\n",
    "print(q_besseli1(0.5, 0.2, 2))\n",
    "print(q_besseli1(0.5+2j, 0.2, 2))\n",
    "print(q_besseli1(0.5, 0.2+2j, 2))\n",
    "print(q_besseli2(0.5, 0.2, 0))\n",
    "print(q_besseli2(0.5, 0.2, 1))\n",
    "print(q_besseli2(0.5, 0.2, 2))\n",
    "print(q_besseli2(0.5+2j, 0.2, 2))\n",
    "print(q_besseli2(0.5, 0.2+2j, 2))\n",
    "print(q_besseli3(0.5, 0.2, 0))\n",
    "print(q_besseli3(0.5, 0.2, 1))\n",
    "print(q_besseli3(0.5, 0.2, 2))\n",
    "print(q_besseli3(0.5+2j, 0.2, 2))\n",
    "print(q_besseli3(0.5, 0.2+2j, 2))\n",
    "print(q_besselj1p(0.5, 0.2, 0))\n",
    "print(q_besselj1p(0.5, 0.2, 1))\n",
    "print(q_besselj1p(0.5, 0.2, 2))\n",
    "print(q_besselj2p(0.5, 0.2, 0))\n",
    "print(q_besselj2p(0.5, 0.2, 1))\n",
    "print(q_besselj2p(0.5, 0.2, 2))\n",
    "print(q_besselj3p(0.5, 0.2, 0))\n",
    "print(q_besselj3p(0.5, 0.2, 1))\n",
    "print(q_besselj3p(0.5, 0.2, 2))\n",
    "print(q_besseli1p(0.5, 0.2, 0))\n",
    "print(q_besseli1p(0.5, 0.2, 1))\n",
    "print(q_besseli1p(0.5, 0.2, 2))\n",
    "print(q_besseli2p(0.5, 0.2, 0))\n",
    "print(q_besseli2p(0.5, 0.2, 1))\n",
    "print(q_besseli2p(0.5, 0.2, 2))\n",
    "print(q_besseli3p(0.5, 0.2, 0))\n",
    "print(q_besseli3p(0.5, 0.2, 1))\n",
    "print(q_besseli3p(0.5, 0.2, 2))\n",
    "print(q_besselj1p(0.5, 0.2, 0, type=2))\n",
    "print(q_besselj1p(0.5, 0.2, 1, type=2))\n",
    "print(q_besselj1p(0.5, 0.2, 2, type=2))\n",
    "print(q_besselj2p(0.5, 0.2, 0, type=2))\n",
    "print(q_besselj2p(0.5, 0.2, 1, type=2))\n",
    "print(q_besselj2p(0.5, 0.2, 2, type=2))\n",
    "print(q_besselj3p(0.5, 0.2, 0, type=2))\n",
    "print(q_besselj3p(0.5, 0.2, 1, type=2))\n",
    "print(q_besselj3p(0.5, 0.2, 2, type=2))\n",
    "print(q_besseli1p(0.5, 0.2, 0, type=2))\n",
    "print(q_besseli1p(0.5, 0.2, 1, type=2))\n",
    "print(q_besseli1p(0.5, 0.2, 2, type=2))\n",
    "print(q_besseli2p(0.5, 0.2, 0, type=2))\n",
    "print(q_besseli2p(0.5, 0.2, 1, type=2))\n",
    "print(q_besseli2p(0.5, 0.2, 2, type=2))\n",
    "print(q_besseli3p(0.5, 0.2, 0, type=2))\n",
    "print(q_besseli3p(0.5, 0.2, 1, type=2))\n",
    "print(q_besseli3p(0.5, 0.2, 2, type=2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h4>q-exponential functions</h4>\n",
    "<a href=\"https://dlmf.nist.gov/17.3#i\">q-exponential functions</a>\n",
    "$$e_q(x)=\\frac1{((1-q)x;q)_\\infty}\\quad(17.3.1)$$\n",
    "$$E_q(x)=\\frac1{(-(1-q)x;q)_\\infty}\\quad(17.3.2)$$\n",
    "<h4>q-sine functions</h4>\n",
    "<a href=\"https://dlmf.nist.gov/17.3#i\">q-sine functions</a>\n",
    "$$\\sin_q(x)=\\tfrac1{2i}(e_q(ix)-e_q(-ix))\\quad(17.3.3)$$\n",
    "$$Sin_q(x)=\\tfrac1{2i}(E_q(ix)-E_q(-ix))\\quad(17.3.4)$$\n",
    "<h4>q-cosine functions</h4>\n",
    "<a href=\"https://dlmf.nist.gov/17.3#i\">q-cosine functions</a>\n",
    "$$\\cos_q(x)=\\tfrac12(e_q(ix)+e_q(-ix))\\quad(17.3.5)$$\n",
    "$$Cos_q(x)=\\tfrac12(E_q(ix)+E_q(-ix))\\quad(17.3.6)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [],
   "source": [
    "def q_e(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-exponential function e_q is a q-analog of exponential function exp.\"\"\"\n",
    "    return 1/(q_poch(((1 - q)*x), q))\n",
    "\n",
    "\n",
    "def q_E(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-exponential function E_q is a q-analog of exponential function exp.\"\"\"\n",
    "    return 1/(q_poch((-(1 - q)*x), q))\n",
    "\n",
    "\n",
    "def q_sin(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-sin is a q-analog of sine function sin.\"\"\"\n",
    "    result = 1/2j*(q_e(1j*x, q) - q_e(-1j*x, q))\n",
    "    if not result.imag:\n",
    "        return result.real\n",
    "    return result\n",
    "\n",
    "\n",
    "def q_Sin(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-Sin is a q-analog of sine function sin.\"\"\"\n",
    "    result = 1/2j*(q_E(1j*x, q) - q_E(-1j*x, q))\n",
    "    if not result.imag:\n",
    "        return result.real\n",
    "    return result\n",
    "\n",
    "\n",
    "def q_cos(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-cos is a q-analog of cosine function cos.\"\"\"\n",
    "    result = 0.5*(q_e(1j*x, q) + q_e(-1j*x, q))\n",
    "    if not result.imag:\n",
    "        return result.real\n",
    "    return result\n",
    "\n",
    "\n",
    "def q_Cos(x: Numeric, q: Numeric) -> Numeric:\n",
    "    \"\"\"q-Cos is a q-analog of cosine function cos.\"\"\"\n",
    "    result = 0.5*(q_E(1j*x, q) + q_E(-1j*x, q))\n",
    "    if not result.imag:\n",
    "        return result.real\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5943062143614387\n",
      "0.44155612841520503\n",
      "0.3949984325548212\n",
      "-0.5943062143614387\n",
      "-0.44155612841520503\n",
      "-0.3949984325548212\n",
      "0.4665689316001744\n",
      "0.8059671858695627\n",
      "0.8515008534670551\n",
      "0.4665689316001744\n",
      "0.8059671858695627\n",
      "0.8515008534670551\n",
      "-0.022059966374363842\n",
      "0.06336058878373414\n",
      "0.08009974621609586\n",
      "-0.022059966374363842\n",
      "0.06336058878373414\n",
      "0.08009974621609586\n"
     ]
    }
   ],
   "source": [
    "𝜋 = 3.141592653589793\n",
    "print(q_sin(𝜋/3, 0.2))\n",
    "print(q_sin(𝜋/6, 0.2))\n",
    "print(q_sin(𝜋/7, 0.2))\n",
    "print(q_Sin(𝜋/3, 0.2))\n",
    "print(q_Sin(𝜋/6, 0.2))\n",
    "print(q_Sin(𝜋/7, 0.2))\n",
    "print(q_cos(𝜋/3, 0.2))\n",
    "print(q_cos(𝜋/6, 0.2))\n",
    "print(q_cos(𝜋/7, 0.2))\n",
    "print(q_Cos(𝜋/3, 0.2))\n",
    "print(q_Cos(𝜋/6, 0.2))\n",
    "print(q_Cos(𝜋/7, 0.2))\n",
    "print(q_cos(𝜋/2 + 𝜋/3, 0.2))\n",
    "print(q_cos(𝜋/2 + 𝜋/6, 0.2))\n",
    "print(q_cos(𝜋/2 + 𝜋/7, 0.2))\n",
    "print(q_Cos(𝜋/2 + 𝜋/3, 0.2))\n",
    "print(q_Cos(𝜋/2 + 𝜋/6, 0.2))\n",
    "print(q_Cos(𝜋/2 + 𝜋/7, 0.2))"
   ]
  }
 ],
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