(i) sigma^2: s-quadrature vs closed form pi^a Gamma(b)/Gamma(N)
  d=  5 k=  4  quad=2.000000000000e+00  closed=2.000000000000e+00  rel.err=2.2e-16
  d=  6 k=  5  quad=1.570796326795e+00  closed=1.570796326795e+00  rel.err=2.8e-16
  d=  9 k=  6  quad=4.188790204786e+00  closed=4.188790204786e+00  rel.err=2.1e-16
  d= 10 k=  7  quad=2.467401100272e+00  closed=2.467401100272e+00  rel.err=1.8e-16
  d= 15 k=  9  quad=5.167712780050e+00  closed=5.167712780050e+00  rel.err=3.4e-16
  d= 15 k= 12  quad=6.382918407294e-01  closed=6.382918407294e-01  rel.err=5.2e-16
  d= 20 k= 11  quad=1.275082019939e+01  closed=1.275082019939e+01  rel.err=4.2e-16
  d= 21 k= 20  quad=5.990767402532e-01  closed=5.990767402532e-01  rel.err=9.3e-16
  d= 30 k= 17  quad=2.996322646604e-01  closed=2.996322646604e-01  rel.err=1.9e-16
  d= 40 k= 23  quad=6.160941495835e-03  closed=6.160941495835e-03  rel.err=1.7e-15
  d= 60 k= 35  quad=9.107476679304e-07  closed=9.107476675149e-07  rel.err=4.6e-10
(ii) cumulants n=3..6: s-quadrature vs E[X^(n-2)] (Beta) vs paper (6.2)
  d=  5 k=  4 n=3  quad=4.1652027545e-01  beta=4.1652027545e-01  paper=4.1652027545e-01  rel(q,b)=1.3e-16 rel(b,p)=0.0e+00
  d=  5 k=  4 n=4  quad=2.2857142857e-01  beta=2.2857142857e-01  paper=2.2857142857e-01  rel(q,b)=7.3e-16 rel(b,p)=0.0e+00
  d=  5 k=  4 n=6  quad=8.5248085248e-02  beta=8.5248085248e-02  paper=8.5248085248e-02  rel(q,b)=1.6e-16 rel(b,p)=0.0e+00
  d=  6 k=  5 n=3  quad=4.9867785050e-01  beta=4.9867785050e-01  paper=4.9867785050e-01  rel(q,b)=1.1e-16 rel(b,p)=0.0e+00
  d=  6 k=  5 n=4  quad=3.1333629421e-01  beta=3.1333629421e-01  paper=3.1333629421e-01  rel(q,b)=3.5e-16 rel(b,p)=0.0e+00
  d=  6 k=  5 n=6  quad=1.5033679043e-01  beta=1.5033679043e-01  paper=1.5033679043e-01  rel(q,b)=1.1e-15 rel(b,p)=0.0e+00
  d=  9 k=  6 n=3  quad=8.9940823941e-02  beta=8.9940823941e-02  paper=8.9940823941e-02  rel(q,b)=6.8e-15 rel(b,p)=0.0e+00
  d=  9 k=  6 n=4  quad=2.0351481235e-02  beta=2.0351481235e-02  paper=2.0351481235e-02  rel(q,b)=1.0e-15 rel(b,p)=0.0e+00
  d=  9 k=  6 n=6  quad=2.0090847980e-03  beta=2.0090847980e-03  paper=2.0090847980e-03  rel(q,b)=3.7e-15 rel(b,p)=1.9e-15
  d= 10 k=  7 n=3  quad=1.3926057521e-01  beta=1.3926057521e-01  paper=1.3926057521e-01  rel(q,b)=6.0e-16 rel(b,p)=0.0e+00
  d= 10 k=  7 n=4  quad=4.2448034944e-02  beta=4.2448034944e-02  paper=4.2448034944e-02  rel(q,b)=1.6e-16 rel(b,p)=8.2e-16
  d= 10 k=  7 n=6  quad=7.2732908411e-03  beta=7.2732908411e-03  paper=7.2732908411e-03  rel(q,b)=1.1e-15 rel(b,p)=0.0e+00
  d= 15 k=  9 n=3  quad=1.2568480388e-02  beta=1.2568480388e-02  paper=1.2568480388e-02  rel(q,b)=1.4e-16 rel(b,p)=0.0e+00
  d= 15 k=  9 n=4  quad=1.1727830703e-03  beta=1.1727830703e-03  paper=1.1727830703e-03  rel(q,b)=0.0e+00 rel(b,p)=0.0e+00
  d= 15 k=  9 n=6  quad=3.8643769906e-05  beta=3.8643769906e-05  paper=3.8643769906e-05  rel(q,b)=7.9e-14 rel(b,p)=0.0e+00
  d= 15 k= 12 n=3  quad=3.5895981989e-01  beta=3.5895981989e-01  paper=3.5895981989e-01  rel(q,b)=3.1e-16 rel(b,p)=0.0e+00
  d= 15 k= 12 n=4  quad=2.2957645578e-01  beta=2.2957645578e-01  paper=2.2957645578e-01  rel(q,b)=7.3e-16 rel(b,p)=3.5e-15
  d= 15 k= 12 n=6  quad=1.5922901482e-01  beta=1.5922901482e-01  paper=1.5922901482e-01  rel(q,b)=5.4e-15 rel(b,p)=0.0e+00
  d= 20 k= 11 n=3  quad=5.4696681509e-04  beta=5.4696681509e-04  paper=5.4696681509e-04  rel(q,b)=9.9e-16 rel(b,p)=0.0e+00
  d= 20 k= 11 n=4  quad=1.3804683010e-05  beta=1.3804683010e-05  paper=1.3804683010e-05  rel(q,b)=2.3e-15 rel(b,p)=0.0e+00
  d= 20 k= 11 n=6  quad=7.2568801857e-08  beta=7.2568801857e-08  paper=7.2568801857e-08  rel(q,b)=3.2e-12 rel(b,p)=0.0e+00
  d= 21 k= 20 n=3  quad=8.9474535071e-01  beta=8.9474535071e-01  paper=8.9474535071e-01  rel(q,b)=1.4e-15 rel(b,p)=0.0e+00
  d= 21 k= 20 n=4  quad=9.3749663229e-01  beta=9.3749663229e-01  paper=9.3749663229e-01  rel(q,b)=4.1e-15 rel(b,p)=0.0e+00
  d= 21 k= 20 n=6  quad=1.2056844232e+00  beta=1.2056844232e+00  paper=1.2056844232e+00  rel(q,b)=2.9e-14 rel(b,p)=2.9e-14
  d= 30 k= 17 n=3  quad=9.4777379789e-04  beta=9.4777379789e-04  paper=9.4777379789e-04  rel(q,b)=2.3e-15 rel(b,p)=0.0e+00
  d= 30 k= 17 n=4  quad=6.2871491491e-05  beta=6.2871491491e-05  paper=6.2871491491e-05  rel(q,b)=1.1e-15 rel(b,p)=0.0e+00
  d= 30 k= 17 n=6  quad=4.6763508849e-06  beta=4.6763508849e-06  paper=4.6763508849e-06  rel(q,b)=5.1e-15 rel(b,p)=1.4e-14
  d= 40 k= 23 n=3  quad=1.1643757143e-03  beta=1.1643757143e-03  paper=1.1643757143e-03  rel(q,b)=7.1e-15 rel(b,p)=0.0e+00
  d= 40 k= 23 n=4  quad=2.1215157667e-04  beta=2.1215157667e-04  paper=2.1215157667e-04  rel(q,b)=1.4e-14 rel(b,p)=1.4e-14
  d= 40 k= 23 n=6  quad=2.5207239871e-04  beta=2.5207239871e-04  paper=2.5207239871e-04  rel(q,b)=1.0e-13 rel(b,p)=0.0e+00
  d= 60 k= 35 n=3  quad=2.4056921018e-03  beta=2.4056921018e-03  paper=2.4056921018e-03  rel(q,b)=3.6e-16 rel(b,p)=0.0e+00
  d= 60 k= 35 n=4  quad=5.4916457364e-03  beta=5.4916457364e-03  paper=5.4916457364e-03  rel(q,b)=1.7e-14 rel(b,p)=0.0e+00
  d= 60 k= 35 n=6  quad=4.7137971414e+00  beta=4.7137971414e+00  paper=4.7137971414e+00  rel(q,b)=0.0e+00 rel(b,p)=5.7e-14
(iii) log psi*(t): s-quadrature (definition) vs E[g_t(X)] (Beta representation)
  d=  5 k=  4 t=0.5  def=-0.1244066085-0.0086419988i  beta=-0.1244066085-0.0086419988i  |diff|=4.0e-17
  d=  5 k=  4 t=1.0  def=-0.4905937151-0.0682918788i  beta=-0.4905937151-0.0682918788i  |diff|=8.3e-17
  d=  5 k=  4 t=2.0  def=-1.8549757562-0.5202673366i  beta=-1.8549757562-0.5202673366i  |diff|=8.1e-16
  d=  5 k=  4 t=4.0  def=-5.9934632146-3.4380673468i  beta=-5.9934632146-3.4380673468i  |diff|=2.4e-15
  d=  6 k=  5 t=0.5  def=-0.1241872748-0.0103338728i  beta=-0.1241872748-0.0103338728i  |diff|=3.6e-17
  d=  6 k=  5 t=1.0  def=-0.4871511509-0.0813610368i  beta=-0.4871511509-0.0813610368i  |diff|=1.2e-16
  d=  6 k=  5 t=2.0  def=-1.8039770119-0.6108366621i  beta=-1.8039770119-0.6108366621i  |diff|=5.6e-16
  d=  6 k=  5 t=4.0  def=-5.3950479919-3.8172864897i  beta=-5.3950479919-3.8172864897i  |diff|=2.0e-15
  d=  9 k=  6 t=0.5  def=-0.1249470449-0.0018722066i  beta=-0.1249470449-0.0018722066i  |diff|=1.2e-16
  d=  9 k=  6 t=1.0  def=-0.4991548052-0.0149403061i  beta=-0.4991548052-0.0149403061i  |diff|=5.8e-16
  d=  9 k=  6 t=2.0  def=-1.9866091899-0.1183401916i  beta=-1.9866091899-0.1183401916i  |diff|=1.6e-15
  d=  9 k=  6 t=4.0  def=-7.7939105395-0.9104787689i  beta=-7.7939105395-0.9104787689i  |diff|=5.8e-15
  d= 10 k=  7 t=0.5  def=-0.1248896159-0.0028969623i  beta=-0.1248896159-0.0028969623i  |diff|=6.3e-17
  d= 10 k=  7 t=1.0  def=-0.4982413913-0.0230730160i  beta=-0.4982413913-0.0230730160i  |diff|=1.0e-17
  d= 10 k=  7 t=2.0  def=-1.9723370853-0.1813586471i  beta=-1.9723370853-0.1813586471i  |diff|=9.4e-16
  d= 10 k=  7 t=4.0  def=-7.5859451965-1.3549617685i  beta=-7.5859451965-1.3549617685i  |diff|=6.9e-15
  d= 15 k=  9 t=0.5  def=-0.1249969467-0.0002617946i  beta=-0.1249969467-0.0002617946i  |diff|=1.4e-15
  d= 15 k=  9 t=1.0  def=-0.4999511876-0.0020931895i  beta=-0.4999511876-0.0020931895i  |diff|=3.4e-15
  d= 15 k=  9 t=2.0  def=-1.9992215640-0.0167083195i  beta=-1.9992215640-0.0167083195i  |diff|=8.2e-15
  d= 15 k=  9 t=4.0  def=-7.9877061886-0.1324970553i  beta=-7.9877061886-0.1324970553i  |diff|=6.1e-14
  d= 15 k= 12 t=0.5  def=-0.1244055859-0.0074314202i  beta=-0.1244055859-0.0074314202i  |diff|=2.6e-16
  d= 15 k= 12 t=1.0  def=-0.4906518488-0.0583474476i  beta=-0.4906518488-0.0583474476i  |diff|=5.1e-16
  d= 15 k= 12 t=2.0  def=-1.8602085491-0.4339422574i  beta=-1.8602085491-0.4339422574i  |diff|=5.6e-16
  d= 15 k= 12 t=4.0  def=-6.2567814251-2.6783066492i  beta=-6.2567814251-2.6783066492i  |diff|=3.2e-15
  d= 20 k= 11 t=0.5  def=-0.1249999641-0.0000113949i  beta=-0.1249999641-0.0000113949i  |diff|=1.3e-12
  d= 20 k= 11 t=1.0  def=-0.4999994249-0.0000911543i  beta=-0.4999994249-0.0000911543i  |diff|=5.2e-12
  d= 20 k= 11 t=2.0  def=-1.9999908033-0.0007290710i  beta=-1.9999908033-0.0007290710i  |diff|=2.1e-11
  d= 20 k= 11 t=4.0  def=-7.9998531612-0.0058273537i  beta=-7.9998531611-0.0058273537i  |diff|=8.3e-11
  d= 21 k= 20 t=0.5  def=-0.1225846038-0.0183704674i  beta=-0.1225846038-0.0183704674i  |diff|=1.2e-16
  d= 21 k= 20 t=1.0  def=-0.4625707922-0.1406883884i  beta=-0.4625707922-0.1406883884i  |diff|=4.5e-16
  d= 21 k= 20 t=2.0  def=-1.4720722080-0.9475342760i  beta=-1.4720722080-0.9475342760i  |diff|=1.7e-15
  d= 21 k= 20 t=4.0  def=-2.7058842686-4.0352010136i  beta=-2.7058842686-4.0352010136i  |diff|=3.2e-15
  d= 30 k= 17 t=0.5  def=-0.1249998364-0.0000197419i  beta=-0.1249998364-0.0000197419i  |diff|=3.3e-16
  d= 30 k= 17 t=1.0  def=-0.4999973868-0.0001578543i  beta=-0.4999973868-0.0001578543i  |diff|=1.6e-15
  d= 30 k= 17 t=2.0  def=-1.9999584924-0.0012602852i  beta=-1.9999584924-0.0012602852i  |diff|=1.0e-14
  d= 30 k= 17 t=4.0  def=-7.9993538500-0.0100054096i  beta=-7.9993538500-0.0100054096i  |diff|=2.9e-14
  d= 40 k= 23 t=0.5  def=-0.1249994528-0.0000242164i  beta=-0.1249994528-0.0000242164i  |diff|=3.1e-16
  d= 40 k= 23 t=1.0  def=-0.4999914715-0.0001928093i  beta=-0.4999914715-0.0001928093i  |diff|=4.1e-16
  d= 40 k= 23 t=2.0  def=-1.9998740473-0.0015186214i  beta=-1.9998740473-0.0015186214i  |diff|=3.9e-15
  d= 40 k= 23 t=4.0  def=-7.9983231251-0.0116663533i  beta=-7.9983231251-0.0116663533i  |diff|=1.7e-14
  d= 60 k= 35 t=0.5  def=-0.1249929239-0.0000453480i  beta=-0.1249929239-0.0000453480i  |diff|=2.9e-15
  d= 60 k= 35 t=1.0  def=-0.4999254453-0.0003323508i  beta=-0.4999254453-0.0003323508i  |diff|=9.1e-14
  d= 60 k= 35 t=2.0  def=-1.9992900388-0.0023388335i  beta=-1.9992900388-0.0023388335i  |diff|=1.3e-13
  d= 60 k= 35 t=4.0  def=-7.9938319291-0.0157499037i  beta=-7.9938319291-0.0157499037i  |diff|=3.7e-12
worst |diff| in (iii): 8.32e-11
