{
  "index": "1997-A-6",
  "type": "ALG",
  "tag": [
    "ALG",
    "COMB",
    "NT"
  ],
  "difficulty": "",
  "question": "For a positive integer $n$ and any real number $c$, define\n$x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\\geq 0$,\n\\[x_{k+2}=\\frac{cx_{k+1}-(n-k)x_k}{k+1}.\\]\nFix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$.\nFind $x_k$ in terms of $n$ and $k$, $1\\leq k\\leq n$.",
  "solution": "Clearly $x_{n+1}$ is a polynomial in $c$ of degree $n$,\nso it suffices to identify $n$ values of $c$ for which $x_{n+1} =\n0$. We claim these are $c = n-1-2r$ for $r=0,1,\\dots, n-1$; in this\ncase, $x_k$ is the coefficient of $t^{k-1}$ in the polynomial\n$f(t) = (1-t)^r (1+t)^{n-1-r}$. This can be verified by noticing that\n$f$ satisfies the differential equation\n\\[\n\\frac{f'(t)}{f(t)} = \\frac{n-1-r}{1+t} - \\frac{r}{1-t}\n\\]\n(by logarithmic differentiation) or equivalently,\n\\begin{align*}\n(1-t^2) f'(t) &= f(t) [(n-1-r)(1-t) - r(1+t)] \\\\\n&= f(t) [(n-1-2r)  - (n-1)t]\n\\end{align*}\nand then taking the coefficient of $t^{k}$ on both sides:\n\\begin{gather*}\n(k+1) x_{k+2} - (k-1) x_k = \\\\\n(n-1-2r) x_{k+1} - (n-1) x_{k}.\n\\end{gather*}\nIn particular, the largest such $c$ is $n-1$, and $x_k =\n\\binom{n-1}{k-1}$ for $k= 1, 2, \\dots, n$.\n\nGreg Kuperberg has suggested an alternate approach to show directly\nthat $c=n-1$ is the largest root, without computing the others. Note\nthat the condition $x_{n+1} = 0$ states that $(x_1, \\dots, x_n)$ is an\neigenvector of the matrix\n\\[\nA_{ij} = \\left\\{ \\begin{array}{cc} i & j = i + 1 \\\\ n-j & j=i-1 \\\\\n0&\\mbox{otherwise} \\end{array} \\right.\n\\]\nwith eigenvalue $c$. By the Perron-Frobenius theorem, $A$ has a unique\neigenvector with positive entries, whose eigenvalue has modulus\ngreater than or equal to that of any other eigenvalue, which proves\nthe claim.",
  "vars": [
    "k",
    "t",
    "x_k",
    "x_0",
    "x_1",
    "x_k+1",
    "x_k+2",
    "x_n+1",
    "x_n",
    "i",
    "j"
  ],
  "params": [
    "n",
    "c",
    "r",
    "f",
    "A_ij"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "k": "indexer",
        "t": "auxvar",
        "x_k": "indexedvalue",
        "x_0": "zerothvalue",
        "x_1": "firstvalue",
        "x_k+1": "successor",
        "x_k+2": "secondnext",
        "x_n+1": "afterlast",
        "x_n": "lastvalue",
        "i": "rowindex",
        "j": "colindex",
        "n": "sizeparam",
        "c": "coeffparam",
        "r": "rootindex",
        "f": "polyfunc",
        "A_ij": "matrixentry"
      },
      "question": "For a positive integer $sizeparam$ and any real number $coeffparam$, define\n$indexedvalue$ recursively by $zerothvalue=0$, $firstvalue=1$, and for $indexer\\geq 0$,\n\\[secondnext=\\frac{coeffparam\\,successor-(sizeparam-indexer)indexedvalue}{indexer+1}.\\]\nFix $sizeparam$ and then take $coeffparam$ to be the largest value for which $afterlast=0$.\nFind $indexedvalue$ in terms of $sizeparam$ and $indexer$, $1\\leq indexer\\leq sizeparam$.",
      "solution": "Clearly $afterlast$ is a polynomial in $coeffparam$ of degree $sizeparam$,\\nso it suffices to identify $sizeparam$ values of $coeffparam$ for which $afterlast =\\n0$. We claim these are $coeffparam = sizeparam-1-2rootindex$ for $rootindex=0,1,\\dots, sizeparam-1$; in this\\ncase, $indexedvalue$ is the coefficient of $auxvar^{indexer-1}$ in the polynomial\\n$polyfunc(auxvar) = (1-auxvar)^{rootindex} (1+auxvar)^{sizeparam-1-rootindex}$. This can be verified by noticing that\\n$polyfunc$ satisfies the differential equation\\n\\[\\frac{polyfunc'(auxvar)}{polyfunc(auxvar)} = \\frac{sizeparam-1-rootindex}{1+auxvar} - \\frac{rootindex}{1-auxvar}\\]\\n(by logarithmic differentiation) or equivalently,\\n\\begin{align*}\\n(1-auxvar^2) polyfunc'(auxvar) &= polyfunc(auxvar) [(sizeparam-1-rootindex)(1-auxvar) - rootindex(1+auxvar)] \\\\&= polyfunc(auxvar) [(sizeparam-1-2rootindex)  - (sizeparam-1)auxvar]\\n\\end{align*}\\nand then taking the coefficient of $auxvar^{indexer}$ on both sides:\\n\\begin{gather*}\\n(indexer+1) secondnext - (indexer-1) indexedvalue = \\\\ (sizeparam-1-2rootindex) successor - (sizeparam-1) indexedvalue.\\n\\end{gather*}\\nIn particular, the largest such $coeffparam$ is $sizeparam-1$, and $indexedvalue =\\n\\binom{sizeparam-1}{indexer-1}$ for $indexer= 1, 2, \\dots, sizeparam$.\\n\\nGreg Kuperberg has suggested an alternate approach to show directly\\nthat $coeffparam=sizeparam-1$ is the largest root, without computing the others. Note\\nthat the condition $afterlast = 0$ states that $(firstvalue, \\dots, lastvalue)$ is an\\neigenvector of the matrix\\n\\[matrixentry = \\left\\{ \\begin{array}{cc} rowindex & colindex = rowindex + 1 \\\\ sizeparam-colindex & colindex=rowindex-1 \\\\ 0&\\mbox{otherwise} \\end{array} \\right.\\]\\nwith eigenvalue $coeffparam$. By the Perron-Frobenius theorem, $A$ has a unique\\neigenvector with positive entries, whose eigenvalue has modulus\\ngreater than or equal to that of any other eigenvalue, which proves\\nthe claim."
    },
    "descriptive_long_confusing": {
      "map": {
        "k": "driftwood",
        "t": "pineapple",
        "x_k": "fireplace",
        "x_0": "trebuchet",
        "x_1": "starfruit",
        "x_k+1": "motorcycle",
        "x_k+2": "sugarcane",
        "x_n+1": "blacksmith",
        "x_n": "lighthouse",
        "i": "waterfall",
        "j": "afterglow",
        "n": "passenger",
        "c": "moonshine",
        "r": "woodpecker",
        "f": "buttercup",
        "A_ij": "windflower"
      },
      "question": "For a positive integer $passenger$ and any real number $moonshine$, define\n$fireplace$ recursively by $trebuchet=0$, $starfruit=1$, and for $driftwood\\geq 0$,\n\\[sugarcane=\\frac{moonshine\\,motorcycle-(passenger-driftwood)fireplace}{driftwood+1}.\\]\nFix $passenger$ and then take $moonshine$ to be the largest value for which $blacksmith=0$.\nFind $fireplace$ in terms of $passenger$ and $driftwood$, $1\\leq driftwood\\leq passenger$.",
      "solution": "Clearly $blacksmith$ is a polynomial in $moonshine$ of degree $passenger$, so it suffices to identify $passenger$ values of $moonshine$ for which $blacksmith = 0$. We claim these are $moonshine = passenger-1-2woodpecker$ for $woodpecker=0,1,\\dots, passenger-1$; in this case, $fireplace$ is the coefficient of $pineapple^{driftwood-1}$ in the polynomial\n$buttercup(pineapple) = (1-pineapple)^{woodpecker} (1+pineapple)^{passenger-1-woodpecker}$. This can be verified by noticing that $buttercup$ satisfies the differential equation\n\\[\\frac{buttercup'(pineapple)}{buttercup(pineapple)} = \\frac{passenger-1-woodpecker}{1+pineapple} - \\frac{woodpecker}{1-pineapple}\\]\n(by logarithmic differentiation) or equivalently,\n\\begin{align*}\n(1-pineapple^2) buttercup'(pineapple) &= buttercup(pineapple) [(passenger-1-woodpecker)(1-pineapple) - woodpecker(1+pineapple)] \\\\ &= buttercup(pineapple) [(passenger-1-2woodpecker)  - (passenger-1)pineapple]\n\\end{align*}\nand then taking the coefficient of $pineapple^{driftwood}$ on both sides:\n\\begin{gather*}\n(driftwood+1) sugarcane - (driftwood-1) fireplace = \\\\ (passenger-1-2woodpecker) motorcycle - (passenger-1) fireplace.\n\\end{gather*}\nIn particular, the largest such $moonshine$ is $passenger-1$, and $fireplace = \\binom{passenger-1}{driftwood-1}$ for $driftwood= 1, 2, \\dots, passenger$.\n\nGreg Kuperberg has suggested an alternate approach to show directly that $moonshine=passenger-1$ is the largest root, without computing the others. Note that the condition $blacksmith = 0$ states that $(starfruit, \\dots, lighthouse)$ is an eigenvector of the matrix\n\\[windflower = \\left\\{ \\begin{array}{cc} waterfall & afterglow = waterfall + 1 \\\\ passenger-afterglow & afterglow=waterfall-1 \\\\ 0&\\mbox{otherwise} \\end{array} \\right.\\]\nwith eigenvalue $moonshine$. By the Perron-Frobenius theorem, $A$ has a unique eigenvector with positive entries, whose eigenvalue has modulus greater than or equal to that of any other eigenvalue, which proves the claim."
    },
    "descriptive_long_misleading": {
      "map": {
        "k": "unaltered",
        "t": "stasisval",
        "x_k": "frozenval",
        "x_0": "frozenzero",
        "x_1": "frozenone",
        "x_k+1": "frozensuc",
        "x_k+2": "frozenduo",
        "x_n+1": "frozenend",
        "x_n": "frozenmax",
        "i": "indexless",
        "j": "pointerless",
        "n": "fluidity",
        "c": "minimums",
        "r": "expander",
        "f": "nullifier",
        "A_ij": "scalarijj"
      },
      "question": "For a positive integer $fluidity$ and any real number $minimums$, define\n$frozenval$ recursively by $frozenzero=0$, $frozenone=1$, and for $unaltered\\geq 0$,\n\\[frozenduo=\\frac{minimumsfrozensuc-(fluidity-unaltered)frozenval}{unaltered+1}.\\]\nFix $fluidity$ and then take $minimums$ to be the largest value for which $frozenend=0$.\nFind $frozenval$ in terms of $fluidity$ and $unaltered$, $1\\leq unaltered\\leq fluidity$.",
      "solution": "Clearly $frozenend$ is a polynomial in $minimums$ of degree $fluidity$,\nso it suffices to identify $fluidity$ values of $minimums$ for which $frozenend =\n0$. We claim these are $minimums = fluidity-1-2expander$ for $expander=0,1,\\dots, fluidity-1$; in this\ncase, $frozenval$ is the coefficient of $stasisval^{unaltered-1}$ in the polynomial\n$nullifier(stasisval) = (1-stasisval)^{expander} (1+stasisval)^{fluidity-1-expander}$. This can be verified by noticing that\n$nullifier$ satisfies the differential equation\n\\[\n\\frac{nullifier'(stasisval)}{nullifier(stasisval)} = \\frac{fluidity-1-expander}{1+stasisval} - \\frac{expander}{1-stasisval}\n\\]\n(by logarithmic differentiation) or equivalently,\n\\begin{align*}\n(1-stasisval^2) nullifier'(stasisval) &= nullifier(stasisval) [(fluidity-1-expander)(1-stasisval) - expander(1+stasisval)] \\\n&= nullifier(stasisval) [(fluidity-1-2expander)  - (fluidity-1)stasisval]\n\\end{align*}\nand then taking the coefficient of $stasisval^{unaltered}$ on both sides:\n\\begin{gather*}\n(unaltered+1) frozenduo - (unaltered-1) frozenval = \\\\ (fluidity-1-2expander) frozensuc - (fluidity-1) frozenval.\n\\end{gather*}\nIn particular, the largest such $minimums$ is $fluidity-1$, and $frozenval =\n\\binom{fluidity-1}{unaltered-1}$ for $unaltered= 1, 2, \\dots, fluidity$.\n\nGreg Kuperberg has suggested an alternate approach to show directly\nthat $minimums=fluidity-1$ is the largest root, without computing the others. Note\nthat the condition $frozenend = 0$ states that $(frozenone, \\dots, frozenmax)$ is an\neigenvector of the matrix\n\\[\nscalarijj = \\left\\{ \\begin{array}{cc} indexless & pointerless = indexless + 1 \\\\ fluidity-pointerless & pointerless=indexless-1 \\\\ 0&\\mbox{otherwise} \\end{array} \\right.\n\\]\nwith eigenvalue $minimums$. By the Perron-Frobenius theorem, scalarijj has a unique\neigenvector with positive entries, whose eigenvalue has modulus\ngreater than or equal to that of any other eigenvalue, which proves\nthe claim."
    },
    "garbled_string": {
      "map": {
        "k": "pujzrnep",
        "t": "qzxwtygh",
        "x_k": "hbqmrpsa",
        "x_0": "vnjtklsa",
        "x_1": "kzphqwre",
        "x_k+1": "wrefgods",
        "x_{k+1}": "wrefgods",
        "x_k+2": "yvbnsicu",
        "x_{k+2}": "yvbnsicu",
        "x_n+1": "bxlqumye",
        "x_{n+1}": "bxlqumye",
        "x_n": "zlidrofe",
        "x_{n}": "zlidrofe",
        "i": "rdvkejsm",
        "j": "plixowga",
        "n": "asctevud",
        "c": "rpnlyigte",
        "r": "nqxdufio",
        "f": "obrkepma",
        "A_ij": "lkrnpqwf",
        "A_{ij}": "lkrnpqwf"
      },
      "question": "For a positive integer $asctevud$ and any real number $rpnlyigte$, define\n$hbqmrpsa$ recursively by $vnjtklsa=0$, $kzphqwre=1$, and for $pujzrnep\\geq 0$,\n\\[\nyvbnsicu=\\frac{rpnlyigte\\,wrefgods-(asctevud-pujzrnep)hbqmrpsa}{pujzrnep+1}.\n\\]\nFix $asctevud$ and then take $rpnlyigte$ to be the largest value for which $bxlqumye=0$.\nFind $hbqmrpsa$ in terms of $asctevud$ and $pujzrnep$, $1\\leq pujzrnep\\leq asctevud$.",
      "solution": "Clearly $bxlqumye$ is a polynomial in $rpnlyigte$ of degree $asctevud$,\nso it suffices to identify $asctevud$ values of $rpnlyigte$ for which $bxlqumye =\n0$. We claim these are $rpnlyigte = asctevud-1-2nqxdufio$ for $nqxdufio=0,1,\\dots, asctevud-1$; in this\ncase, $hbqmrpsa$ is the coefficient of $qzxwtygh^{pujzrnep-1}$ in the polynomial\n$obrkepma(qzxwtygh) = (1-qzxwtygh)^{nqxdufio} (1+qzxwtygh)^{asctevud-1-nqxdufio}$. This can be verified by noticing that\n$obrkepma$ satisfies the differential equation\n\\[\n\\frac{obrkepma'(qzxwtygh)}{obrkepma(qzxwtygh)} = \\frac{asctevud-1-nqxdufio}{1+qzxwtygh} - \\frac{nqxdufio}{1-qzxwtygh}\n\\]\n(by logarithmic differentiation) or equivalently,\n\\begin{align*}\n(1-qzxwtygh^2)\\, obrkepma'(qzxwtygh) &= obrkepma(qzxwtygh) [(asctevud-1-nqxdufio)(1-qzxwtygh) - nqxdufio(1+qzxwtygh)] \\\\\n&= obrkepma(qzxwtygh) [(asctevud-1-2nqxdufio)  - (asctevud-1)qzxwtygh]\n\\end{align*}\nand then taking the coefficient of $qzxwtygh^{pujzrnep}$ on both sides:\n\\begin{gather*}\n(pujzrnep+1)\\, yvbnsicu - (pujzrnep-1)\\, hbqmrpsa = \\\\\n(asctevud-1-2nqxdufio)\\, wrefgods - (asctevud-1)\\, hbqmrpsa.\n\\end{gather*}\nIn particular, the largest such $rpnlyigte$ is $asctevud-1$, and $hbqmrpsa =\n\\binom{asctevud-1}{pujzrnep-1}$ for $pujzrnep= 1, 2, \\dots, asctevud$.\n\nGreg Kuperberg has suggested an alternate approach to show directly\nthat $rpnlyigte=asctevud-1$ is the largest root, without computing the others. Note\nthat the condition $bxlqumye = 0$ states that $(kzphqwre, \\dots, zlidrofe)$ is an\neigenvector of the matrix\n\\[\nlkrnpqwf = \\left\\{ \\begin{array}{cc} rdvkejsm & plixowga = rdvkejsm + 1 \\\\ asctevud-plixowga & plixowga=rdvkejsm-1 \\\\\n0&\\mbox{otherwise} \\end{array} \\right.\n\\]\nwith eigenvalue $rpnlyigte$. By the Perron-Frobenius theorem, $lkrnpqwf$ has a unique\neigenvector with positive entries, whose eigenvalue has modulus\ngreater than or equal to that of any other eigenvalue, which proves\nthe claim."
    },
    "kernel_variant": {
      "question": "Fix an integer                                                   n ≥ 2\nand a real number                                                q>0 , q≠1.\nFor every real parameter c consider the sequence (x_k)_{k≥0} defined by  \n\n  x₀=0 , x₁=1 , and for k=0,1,…  \n\n    x_{k+2}=  \\dfrac{c\\,x_{k+1}-(n-k)\\,x_{k}}{[\\,k+1\\,]_{q}},    (★)\n\nwhere the q–integer  \n\n    [\\,m\\,]_{q}:=\\dfrac{1-q^{m}}{1-q}=1+q+\\dots +q^{m-1}\\qquad(m∈ℕ).\n\nPut  \n\n    c_n(q):=\\max\\{\\,c>0\\mid x_{n+1}(c)=0\\,\\}.    (∗)\n\n1.  Show the equivalence  \n\n   x_{n+1}(c)=0 ⇔ P_n(c)=0,\n\nwhere P₀(λ):=1, P₁(λ):=λ, and for 1≤k≤n−1  \n\n    P_{k+1}(λ)=λP_k(λ)-(n-k)[k]_q\\,P_{k-1}(λ).    (1)\n\nProve that P_n possesses n distinct real zeros and that c_n(q) is the\nlargest one.\n\n2.  Let J_n(q) be the n×n symmetric Jacobi matrix  \n\n  (J_n(q))_{k,k}=0 (1≤k≤n),  \n  (J_n(q))_{k,k+1}=(J_n(q))_{k+1,k}=√{(n-k)[k]_q} (1≤k≤n−1).\n\nProve that P_n is the characteristic polynomial of J_n(q) and hence  \n\n    c_n(q)=ρ(J_n(q)), \n\nthe spectral radius of J_n(q).  (Consequently c_n(q) is algebraic of\ndegree at most n.)\n\n3.  For c=c_n(q) establish the closed formula  \n\n    x_k=\\dfrac{P_{k-1}\\!\\bigl(c_n(q)\\bigr)}\n        {\\,\\prod_{j=1}^{\\,k-1}[j]_q}\\qquad(1≤k≤n)    (2)\n\nand show that x_k>0 for 1≤k≤n.\n\n4.  Compute c_n(q) and the corresponding sequence (x_k)_{k≤n}\nexplicitly for n=2,3,4.",
      "solution": "Throughout we extend the definition of P_k by putting P_{−1}:=0,\nP₀:=1 and using (1) for every k≥0.\n\n────────────────────────────────────────\n1.  Connection formula for (x_k).  \n   We prove by induction on k that  \n\n   x_{k+1}(c)=\\dfrac{P_k(c)}{\\,[1]_q[2]_q⋯[k]_q}\\qquad(k≥0).  (3)\n\n   For k=0,1 it is immediate from the initial data.  \n   Assuming (3) for k and k−1, use (★) and (1):\n\n  x_{k+2}\n   =\\frac{c\\,P_k(c)-(n-k)[k]_q P_{k-1}(c)}\n          {[1]_q⋯[k]_q[k+1]_q}\n   =\\frac{P_{k+1}(c)}{[1]_q⋯[k]_q[k+1]_q}.\n\n   Hence (3) is true for every k; in particular  \n\n   x_{n+1}(c)=0 ⇔ P_n(c)=0.  (4)\n\n────────────────────────────────────────\n2.  Jacobi matrix, reality and simplicity of the zeros.  \n\n   For 1≤k≤n let J_k(q) be the k×k leading principal sub-matrix of\n   J_n(q) and set Δ_k(λ):=det(λI_k−J_k(q)), Δ₀(λ):=1.\n   A Laplace expansion along the last row/column gives\n\n  Δ_{k+1}(λ)=λΔ_k(λ)-(n-k)[k]_q Δ_{k-1}(λ) (1≤k≤n−1).\n\n   Because Δ₀=P₀ and Δ₁=P₁, comparison with (1) yields  \n\n    Δ_k=P_k (0≤k≤n).  (5)\n\n   Hence P_n is the characteristic polynomial of the real symmetric\n   matrix J_n(q).  Such a matrix is diagonalizable with real\n   eigenvalues.  Moreover J_n(q) is irreducible tridiagonal with strictly\n   positive off-diagonal entries; therefore all its eigenvalues are\n   simple.  By (5) the n zeros of P_n are exactly these n simple real\n   eigenvalues, which we order as  \n\n   λ_{n,1}<λ_{n,2}<⋯<λ_{n,n}.  (6)\n\n   The largest one is  \n\n   c_n(q):=λ_{n,n}=ρ(J_n(q)).  (7)\n\n────────────────────────────────────────\n3.  Closed formula and positivity of the extremal sequence.  \n\n   Formula (2) follows immediately from (3) once c=c_n(q) is chosen.\n\n   We now prove x_k>0 (1≤k≤n).  Put λ:=c_n(q)=λ_{n,n}.\n   Because λ is the largest eigenvalue, the matrix λI_k−J_k(q) is\n   positive definite for every k<n; consequently\n\n   P_k(λ)=det(λI_k−J_k(q))>0 (0≤k≤n−1).  (8)\n\n   Insert (8) into (2): every numerator and every denominator is\n   positive, so x_k>0.  No eigenvector or Perron–Frobenius argument is\n   required.\n\n────────────────────────────────────────\n4.  Explicit evaluation for n=2,3,4.  \n   Recall [1]_q=1, [2]_q=1+q, [3]_q=1+q+q².\n\n   • n=2.  \n     P₂(λ)=λ²−1; c₂(q)=1.  \n\n   x₁=1,\\qquad x₂=1.\n\n   • n=3.  \n     P₃(λ)=λ³−(3+q)λ; c₃(q)=√{3+q}.  \n\n   x₁=1,\\qquad x₂=√{3+q},\\qquad  \n   x₃=\\dfrac{c₃(q)²-2}{1+q}=1.\n\n   • n=4.  \n     Using (1) one finds  \n\n   P₃(λ)=λ³-(5+2q)λ,  \n   P₄(λ)=λ⁴-(6+3q+q²)λ²+3(1+q+q²).\n\n     The largest zero of P₄ is  \n\n   c₄(q)=\\sqrt{\\dfrac{6+3q+q^{2}+\\sqrt{(6+3q+q^{2})^{2}-12(1+q+q^{2})}}{2}}.\n\n     Substituting in (2) gives  \n\n   x₁ = 1,\n\n   x₂ = c₄(q),\n\n   x₃ = \\dfrac{c₄(q)^{2}-3}{1+q},\n\n   x₄ = \\dfrac{c₄(q)^{3}-(5+2q)\\,c₄(q)}\n      {(1+q)\\,(1+q+q^{2})}.  (9)\n\n     From (8) each numerator in (9) is positive, so x₄>0, in agreement\n     with the general positivity established above.\n\n   For n≥5 radicals are impossible in general, but (7) together with\n   (2) always produces the extremal sequence after finitely many\n   algebraic operations.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.754564",
        "was_fixed": false,
        "difficulty_analysis": "•  A second independent parameter q is introduced; every integer in the\n   recurrence is replaced by its q-analogue, so ordinary binomial tools\n   no longer suffice—one now needs fluency with Gaussian coefficients,\n   q-calculus and basic hypergeometric functions.\n\n•  The generating-function method still works, but the derivative\n   identity takes the non-trivial q-form (1-q t)F' = … , requiring\n   familiarity with how power-series coefficients behave under q-shifts.\n\n•  Identifying the largest c involves recognizing when an analytic\n   expression (2) collapses to a finite q-binomial polynomial; this is\n   considerably subtler than the linear observation “degree ≤ n−1” used\n   in the classical case.\n\n•  The final formula mixes multiplicative q-powers\n   q^{(k-1)(k-2)/2} with Gaussian coefficients, so even writing the\n   answer demands advanced notation.\n\n•  When q→1 the problem reduces to the original, guaranteeing that the\n   new variant is strictly more general—and thus strictly harder—while\n   preserving the core combinatorial idea."
      }
    },
    "original_kernel_variant": {
      "question": "Fix an integer                                                   n ≥ 2\nand a real number                                                q>0 , q≠1.\nFor every real parameter c consider the sequence (x_k)_{k≥0} defined by  \n\n  x₀=0 , x₁=1 , and for k=0,1,…  \n\n    x_{k+2}=  \\dfrac{c\\,x_{k+1}-(n-k)\\,x_{k}}{[\\,k+1\\,]_{q}},    (★)\n\nwhere the q–integer  \n\n    [\\,m\\,]_{q}:=\\dfrac{1-q^{m}}{1-q}=1+q+\\dots +q^{m-1}\\qquad(m∈ℕ).\n\nPut  \n\n    c_n(q):=\\max\\{\\,c>0\\mid x_{n+1}(c)=0\\,\\}.    (∗)\n\n1.  Show the equivalence  \n\n   x_{n+1}(c)=0 ⇔ P_n(c)=0,\n\nwhere P₀(λ):=1, P₁(λ):=λ, and for 1≤k≤n−1  \n\n    P_{k+1}(λ)=λP_k(λ)-(n-k)[k]_q\\,P_{k-1}(λ).    (1)\n\nProve that P_n possesses n distinct real zeros and that c_n(q) is the\nlargest one.\n\n2.  Let J_n(q) be the n×n symmetric Jacobi matrix  \n\n  (J_n(q))_{k,k}=0 (1≤k≤n),  \n  (J_n(q))_{k,k+1}=(J_n(q))_{k+1,k}=√{(n-k)[k]_q} (1≤k≤n−1).\n\nProve that P_n is the characteristic polynomial of J_n(q) and hence  \n\n    c_n(q)=ρ(J_n(q)), \n\nthe spectral radius of J_n(q).  (Consequently c_n(q) is algebraic of\ndegree at most n.)\n\n3.  For c=c_n(q) establish the closed formula  \n\n    x_k=\\dfrac{P_{k-1}\\!\\bigl(c_n(q)\\bigr)}\n        {\\,\\prod_{j=1}^{\\,k-1}[j]_q}\\qquad(1≤k≤n)    (2)\n\nand show that x_k>0 for 1≤k≤n.\n\n4.  Compute c_n(q) and the corresponding sequence (x_k)_{k≤n}\nexplicitly for n=2,3,4.",
      "solution": "Throughout we extend the definition of P_k by putting P_{−1}:=0,\nP₀:=1 and using (1) for every k≥0.\n\n────────────────────────────────────────\n1.  Connection formula for (x_k).  \n   We prove by induction on k that  \n\n   x_{k+1}(c)=\\dfrac{P_k(c)}{\\,[1]_q[2]_q⋯[k]_q}\\qquad(k≥0).  (3)\n\n   For k=0,1 it is immediate from the initial data.  \n   Assuming (3) for k and k−1, use (★) and (1):\n\n  x_{k+2}\n   =\\frac{c\\,P_k(c)-(n-k)[k]_q P_{k-1}(c)}\n          {[1]_q⋯[k]_q[k+1]_q}\n   =\\frac{P_{k+1}(c)}{[1]_q⋯[k]_q[k+1]_q}.\n\n   Hence (3) is true for every k; in particular  \n\n   x_{n+1}(c)=0 ⇔ P_n(c)=0.  (4)\n\n────────────────────────────────────────\n2.  Jacobi matrix, reality and simplicity of the zeros.  \n\n   For 1≤k≤n let J_k(q) be the k×k leading principal sub-matrix of\n   J_n(q) and set Δ_k(λ):=det(λI_k−J_k(q)), Δ₀(λ):=1.\n   A Laplace expansion along the last row/column gives\n\n  Δ_{k+1}(λ)=λΔ_k(λ)-(n-k)[k]_q Δ_{k-1}(λ) (1≤k≤n−1).\n\n   Because Δ₀=P₀ and Δ₁=P₁, comparison with (1) yields  \n\n    Δ_k=P_k (0≤k≤n).  (5)\n\n   Hence P_n is the characteristic polynomial of the real symmetric\n   matrix J_n(q).  Such a matrix is diagonalizable with real\n   eigenvalues.  Moreover J_n(q) is irreducible tridiagonal with strictly\n   positive off-diagonal entries; therefore all its eigenvalues are\n   simple.  By (5) the n zeros of P_n are exactly these n simple real\n   eigenvalues, which we order as  \n\n   λ_{n,1}<λ_{n,2}<⋯<λ_{n,n}.  (6)\n\n   The largest one is  \n\n   c_n(q):=λ_{n,n}=ρ(J_n(q)).  (7)\n\n────────────────────────────────────────\n3.  Closed formula and positivity of the extremal sequence.  \n\n   Formula (2) follows immediately from (3) once c=c_n(q) is chosen.\n\n   We now prove x_k>0 (1≤k≤n).  Put λ:=c_n(q)=λ_{n,n}.\n   Because λ is the largest eigenvalue, the matrix λI_k−J_k(q) is\n   positive definite for every k<n; consequently\n\n   P_k(λ)=det(λI_k−J_k(q))>0 (0≤k≤n−1).  (8)\n\n   Insert (8) into (2): every numerator and every denominator is\n   positive, so x_k>0.  No eigenvector or Perron–Frobenius argument is\n   required.\n\n────────────────────────────────────────\n4.  Explicit evaluation for n=2,3,4.  \n   Recall [1]_q=1, [2]_q=1+q, [3]_q=1+q+q².\n\n   • n=2.  \n     P₂(λ)=λ²−1; c₂(q)=1.  \n\n   x₁=1,\\qquad x₂=1.\n\n   • n=3.  \n     P₃(λ)=λ³−(3+q)λ; c₃(q)=√{3+q}.  \n\n   x₁=1,\\qquad x₂=√{3+q},\\qquad  \n   x₃=\\dfrac{c₃(q)²-2}{1+q}=1.\n\n   • n=4.  \n     Using (1) one finds  \n\n   P₃(λ)=λ³-(5+2q)λ,  \n   P₄(λ)=λ⁴-(6+3q+q²)λ²+3(1+q+q²).\n\n     The largest zero of P₄ is  \n\n   c₄(q)=\\sqrt{\\dfrac{6+3q+q^{2}+\\sqrt{(6+3q+q^{2})^{2}-12(1+q+q^{2})}}{2}}.\n\n     Substituting in (2) gives  \n\n   x₁ = 1,\n\n   x₂ = c₄(q),\n\n   x₃ = \\dfrac{c₄(q)^{2}-3}{1+q},\n\n   x₄ = \\dfrac{c₄(q)^{3}-(5+2q)\\,c₄(q)}\n      {(1+q)\\,(1+q+q^{2})}.  (9)\n\n     From (8) each numerator in (9) is positive, so x₄>0, in agreement\n     with the general positivity established above.\n\n   For n≥5 radicals are impossible in general, but (7) together with\n   (2) always produces the extremal sequence after finitely many\n   algebraic operations.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.580731",
        "was_fixed": false,
        "difficulty_analysis": "•  A second independent parameter q is introduced; every integer in the\n   recurrence is replaced by its q-analogue, so ordinary binomial tools\n   no longer suffice—one now needs fluency with Gaussian coefficients,\n   q-calculus and basic hypergeometric functions.\n\n•  The generating-function method still works, but the derivative\n   identity takes the non-trivial q-form (1-q t)F' = … , requiring\n   familiarity with how power-series coefficients behave under q-shifts.\n\n•  Identifying the largest c involves recognizing when an analytic\n   expression (2) collapses to a finite q-binomial polynomial; this is\n   considerably subtler than the linear observation “degree ≤ n−1” used\n   in the classical case.\n\n•  The final formula mixes multiplicative q-powers\n   q^{(k-1)(k-2)/2} with Gaussian coefficients, so even writing the\n   answer demands advanced notation.\n\n•  When q→1 the problem reduces to the original, guaranteeing that the\n   new variant is strictly more general—and thus strictly harder—while\n   preserving the core combinatorial idea."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}