## 2 A Necessary but Not Sufficient Condition
### The Heun Equation  
Over the past two decades [PERSON]'s equation has attracted considerable attention due to its increasing number of applications in applied mathematics and theoretical physics [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 82; 85; 87; 89; 91; 86; 88; 87; 88; 89; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 111; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127; 128; 129; 130; 131; 132; 133; 134; 135; 136; 137; 138; 139; 140; 141; 142; 143; 144; 145; 146; 147; 148; 149; 150; 151; 152; 153; 154; 155; 156; 157; 158; 159; 160; 161; 162; 163; 164; 165; 166; 167; 168; 169; 170; 171; 172; 173; 174; 175; 176; 177; 178; 179; 180; 181; 182; 183; 184; 185; 186; 187; 188; 189; 190; 191; 192; 193; 194; 195; 196; 197; 198; 199; 200; 201; 202; 203; 204; 205; 206; 207; 208; 209; 211; 212; 224; 25; 261; 270; 209; 228; 294; 295; 201; 206; 207; 208; 209; 212; 213; 201; 209; 229; 231; 240; 201; 202; 207; 209; 2141; 251; 252; 262; 271; 281; 296; 297; 298; 300; 215; 299; 3160; 20 where the canonical form [41] may be obtained using the substitutions given in Table 1.  
We immediately see that because \(\alpha_{0}=0\) and \(\alpha_{1}\
eq 0\) that (78) falls under Theorem 5. By Equation (53) we obtain the three-term recurrence relation  
\[\left[\alpha_{3}\left(\ell+s-2\right)\left(\ell+s-1\right)+\beta_{2} \left(\ell+s-1\right)-\tau_{1}\right]\mathcal{C}_{\ell-1}+\left[\alpha_{2} \left(\ell+s-1\right)\left(\ell+s\right)+\beta_{1}\left(\ell+s\right)-\tau_{0 }\right]\mathcal{C}_{\ell}\] \[+\left[\alpha_{1}\left(\ell+s\right)\left(\ell+s+1\right)+\beta_ {0}\left(\ell+s+1\right)\right]\mathcal{C}_{\ell+1}=0\,, \tag{79}\]  
for \(\ell=0\), \(1\), \(2\),..., \(m+1\) and \(s\in\left\{0\,,\,1-\left(\beta_{0}/\alpha_{1}\right)\right\}\). We may then input this into the Mathematica(r) program or work directly with the \(\left(n+2\right)\) equations generated by the recurrence relations (79). Using either approach with \(s=0\) will generate the results, as follows. For the zero-degree (\(m=0\)) polynomial solution, \(\mathcal{P}_{0}(r)=1\), \(\ell=0\), \(1\). Noting that \(\mathcal{C}_{j}=0\) for all \(j\
eq 0\), it follows for \(\ell=0\) that \(\tau_{0}=0\), while for \(\ell=1\), \(\tau_{1}=0\). For the first-degree (\(m=1\)) polynomial solution, \(\mathcal{P}_{1}(r)=\mathcal{C}_{0}+\mathcal{C}_{1}\,r\), \(\ell=0\), \(1\), \(2\). It follows that  
\[\ell=0:\quad\mathcal{C}_{1}=\tau_{0}/\beta_{0}\]  
\[\ell=1:\quad\left| \begin{array}{cc}-\tau_{0}&\beta_{0}\\ -\tau_{1}&\beta_{1}-\tau_{0}\end{array} \right|=0\,,\quad\text{(SC)}\]  
\[\ell=2:\quad\tau_{1}=\beta_{2}\,.\quad\text{(NC)} \tag{80}\]  
where (SC) refers to the sufficient condition and (NC) refers to the necessary condition, and we have set \(\mathcal{C}_{0}=1\) for convenience. For the second-degree (\(m=2\)) polynomial solution \(\mathcal{P}_{2}(r)=1+\mathcal{C}_{1}\,r+\mathcal{C}_{2}\,r^{2}\), \(\ell=0\), \(1\), \(2\), \(3\) yields  
\[\ell=0,\,1:\quad\left\{ \begin{array}{cc}\mathcal{C}_{1}=\left| \begin{array}{cc}\tau_{0}&0\\ \tau_{1}&2\left(\alpha_{1}+\beta_{0}\right)\\ \hline\beta_{0}&0\\ \beta_{1}-\tau_{0}&2\left(\alpha_{1}+\beta_{0}\right)\end{array}\right|,\\ \mathcal{C}_{2}=\left|\begin{array}{cc}\beta_{0}&\tau_{0}\\ \beta_{1}-\tau_{0}&\tau_{1}\\ \hline\beta_{0}&0\\ \beta_{1}-\tau_{0}&2\left(\alpha_{1}+\beta_{0}\right)\end{array}\right|,\\ \ell=2:\quad\left|\begin{array}{cc}-\tau_{0}&\beta_{0}&0\\ -\tau_{1}&\beta_{1}-\tau_{0}&2\left(\alpha_{1}+\beta_{0}\right)\\ 0&\beta_{2}-\tau_{1}&2\left(\alpha_{2}+\beta_{1}\right)-\tau_{0}\end{array} \right|=0\,,\quad\text{(SC)}\\ \ell=3:\quad 2\,\alpha_{3}+2\,\beta_{2}-\tau_{1}=0\,.\quad\text{(NC)}\end{array} \right. \tag{81}\]