Jacobi polynomials
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Polynomials
$\alpha$
$\beta$
$n$
$P_n^{(\alpha,\beta)}(x)$
a
a:
a
b
0:
1
a
b
1:
1/2*a*x + 1/2*b*x + 1/2*a - 1/2*b + x
a
b
2:
1/8*a^2*x^2 + 1/4*a*b*x^2 + 1/8*b^2*x^2 + 1/4*a^2*x - 1/4*b^2*x + 7/8*a*x^2 + 7/8*b*x^2 + 1/8*a^2 - 1/4*a*b + 1/8*b^2 + 3/4*a*x - 3/4*b*x + 3/2*x^2 - 1/8*a - 1/8*b - 1/2
a
b
3:
1/48*a^3*x^3 + 1/16*a^2*b*x^3 + 1/16*a*b^2*x^3 + 1/48*b^3*x^3 + 1/16*a^3*x^2 + 1/16*a^2*b*x^2 - 1/16*a*b^2*x^2 - 1/16*b^3*x^2 + 5/16*a^2*x^3 + 5/8*a*b*x^3 + 5/16*b^2*x^3 + 1/16*a^3*x - 1/16*a^2*b*x - 1/16*a*b^2*x + 1/16*b^3*x + 9/16*a^2*x^2 - 9/16*b^2*x^2 + 37/24*a*x^3 + 37/24*b*x^3 + 1/48*a^3 - 1/16*a^2*b + 1/16*a*b^2 - 1/48*b^3 + 3/16*a^2*x - 5/8*a*b*x + 3/16*b^2*x + 5/4*a*x^2 - 5/4*b*x^2 + 5/2*x^3 - 1/16*a^2 + 1/16*b^2 - 5/8*a*x - 5/8*b*x - 1/3*a + 1/3*b - 3/2*x
a
b
4:
1/384*a^4*x^4 + 1/96*a^3*b*x^4 + 1/64*a^2*b^2*x^4 + 1/96*a*b^3*x^4 + 1/384*b^4*x^4 + 1/96*a^4*x^3 + 1/48*a^3*b*x^3 - 1/48*a*b^3*x^3 - 1/96*b^4*x^3 + 13/192*a^3*x^4 + 13/64*a^2*b*x^4 + 13/64*a*b^2*x^4 + 13/192*b^3*x^4 + 1/64*a^4*x^2 - 1/32*a^2*b^2*x^2 + 1/64*b^4*x^2 + 3/16*a^3*x^3 + 3/16*a^2*b*x^3 - 3/16*a*b^2*x^3 - 3/16*b^3*x^3 + 251/384*a^2*x^4 + 251/192*a*b*x^4 + 251/384*b^2*x^4 + 1/96*a^4*x - 1/48*a^3*b*x + 1/48*a*b^3*x - 1/96*b^4*x + 5/32*a^3*x^2 - 7/32*a^2*b*x^2 - 7/32*a*b^2*x^2 + 5/32*b^3*x^2 + 107/96*a^2*x^3 - 107/96*b^2*x^3 + 533/192*a*x^4 + 533/192*b*x^4 + 1/384*a^4 - 1/96*a^3*b + 1/64*a^2*b^2 - 1/96*a*b^3 + 1/384*b^4 + 1/48*a^3*x - 3/16*a^2*b*x + 3/16*a*b^2*x - 1/48*b^3*x + 11/64*a^2*x^2 - 49/32*a*b*x^2 + 11/64*b^2*x^2 + 35/16*a*x^3 - 35/16*b*x^3 + 35/8*x^4 - 1/64*a^3 + 1/64*a^2*b + 1/64*a*b^2 - 1/64*b^3 - 37/96*a^2*x + 37/96*b^2*x - 59/32*a*x^2 - 59/32*b*x^2 - 37/384*a^2 + 43/192*a*b - 37/384*b^2 - 55/48*a*x + 55/48*b*x - 15/4*x^2 + 7/64*a + 7/64*b + 3/8
a
b
5:
1/3840*a^5*x^5 + 1/768*a^4*b*x^5 + 1/384*a^3*b^2*x^5 + 1/384*a^2*b^3*x^5 + 1/768*a*b^4*x^5 + 1/3840*b^5*x^5 + 1/768*a^5*x^4 + 1/256*a^4*b*x^4 + 1/384*a^3*b^2*x^4 - 1/384*a^2*b^3*x^4 - 1/256*a*b^4*x^4 - 1/768*b^5*x^4 + 1/96*a^4*x^5 + 1/24*a^3*b*x^5 + 1/16*a^2*b^2*x^5 + 1/24*a*b^3*x^5 + 1/96*b^4*x^5 + 1/384*a^5*x^3 + 1/384*a^4*b*x^3 - 1/192*a^3*b^2*x^3 - 1/192*a^2*b^3*x^3 + 1/384*a*b^4*x^3 + 1/384*b^5*x^3 + 5/128*a^4*x^4 + 5/64*a^3*b*x^4 - 5/64*a*b^3*x^4 - 5/128*b^4*x^4 + 127/768*a^3*x^5 + 127/256*a^2*b*x^5 + 127/256*a*b^2*x^5 + 127/768*b^3*x^5 + 1/384*a^5*x^2 - 1/384*a^4*b*x^2 - 1/192*a^3*b^2*x^2 + 1/192*a^2*b^3*x^2 + 1/384*a*b^4*x^2 - 1/384*b^5*x^2 + 5/96*a^4*x^3 - 1/96*a^3*b*x^3 - 1/8*a^2*b^2*x^3 - 1/96*a*b^3*x^3 + 5/96*b^4*x^3 + 335/768*a^3*x^4 + 335/768*a^2*b*x^4 - 335/768*a*b^2*x^4 - 335/768*b^3*x^4 + 125/96*a^2*x^5 + 125/48*a*b*x^5 + 125/96*b^2*x^5 + 1/768*a^5*x - 1/256*a^4*b*x + 1/384*a^3*b^2*x + 1/384*a^2*b^3*x - 1/256*a*b^4*x + 1/768*b^5*x + 5/192*a^4*x^2 - 1/12*a^3*b*x^2 + 1/12*a*b^3*x^2 - 5/192*b^4*x^2 + 115/384*a^3*x^3 - 239/384*a^2*b*x^3 - 239/384*a*b^2*x^3 + 115/384*b^3*x^3 + 275/128*a^2*x^4 - 275/128*b^2*x^4 + 1627/320*a*x^5 + 1627/320*b*x^5 + 1/3840*a^5 - 1/768*a^4*b + 1/384*a^3*b^2 - 1/384*a^2*b^3 + 1/768*a*b^4 - 1/3840*b^5 - 1/32*a^3*b*x + 1/16*a^2*b^2*x - 1/32*a*b^3*x - 25/384*a^3*x^2 - 193/384*a^2*b*x^2 + 193/384*a*b^2*x^2 + 25/384*b^3*x^2 - 5/96*a^2*x^3 - 173/48*a*b*x^3 - 5/96*b^2*x^3 + 63/16*a*x^4 - 63/16*b*x^4 + 63/8*x^5 - 1/384*a^4 + 1/192*a^3*b - 1/192*a*b^3 + 1/384*b^4 - 85/768*a^3*x + 97/768*a^2*b*x + 97/768*a*b^2*x - 85/768*b^3*x - 245/192*a^2*x^2 + 245/192*b^2*x^2 - 449/96*a*x^3 - 449/96*b*x^3 - 13/768*a^3 + 17/256*a^2*b - 17/256*a*b^2 + 13/768*b^3 - 5/16*a^2*x + a*b*x - 5/16*b^2*x - 49/16*a*x^2 + 49/16*b*x^2 - 35/4*x^3 + 25/384*a^2 - 25/384*b^2 + 47/64*a*x + 47/64*b*x + 4/15*a - 4/15*b + 15/8*x
a
-1/2
0:
1
a
-1/2
1:
1/2*a*x + 1/2*a + 3/4*x + 1/4
a
-1/2
2:
1/8*a^2*x^2 + 1/4*a^2*x + 3/4*a*x^2 + 1/8*a^2 + 3/4*a*x + 35/32*x^2 + 5/16*x - 13/32
a
-1/2
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 9/32*a^2*x^3 + 1/16*a^3*x + 17/32*a^2*x^2 + 239/192*a*x^3 + 1/48*a^3 + 7/32*a^2*x + 79/64*a*x^2 + 231/128*x^3 - 1/32*a^2 - 21/64*a*x + 63/128*x^2 - 61/192*a - 147/128*x - 19/128
a
-1/2
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 1/16*a^3*x^4 + 1/64*a^4*x^2 + 17/96*a^3*x^3 + 427/768*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 49/48*a^2*x^3 + 139/64*a*x^4 + 1/384*a^4 + 1/32*a^3*x + 35/128*a^2*x^2 + 823/384*a*x^3 + 6435/2048*x^4 - 1/96*a^3 - 7/24*a^2*x - 145/128*a*x^2 + 429/512*x^3 - 77/768*a^2 - 141/128*a*x - 2871/1024*x^2 + 1/384*a - 243/512*x + 611/2048
a
-1/2
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 5/512*a^4*x^5 + 1/384*a^5*x^3 + 19/512*a^4*x^4 + 223/1536*a^3*x^5 + 1/384*a^5*x^2 + 13/256*a^4*x^3 + 611/1536*a^3*x^4 + 1095/1024*a^2*x^5 + 1/768*a^5*x + 7/256*a^4*x^2 + 233/768*a^3*x^3 + 1977/1024*a^2*x^4 + 239689/61440*a*x^5 + 1/3840*a^5 + 1/512*a^4*x - 19/768*a^3*x^2 + 117/512*a^2*x^3 + 47161/12288*a*x^4 + 46189/8192*x^5 - 1/512*a^4 - 145/1536*a^3*x - 525/512*a^2*x^2 - 18611/6144*a*x^3 + 12155/8192*x^4 - 29/1536*a^3 - 369/1024*a^2*x - 18107/6144*a*x^2 - 26455/4096*x^3 + 33/1024*a^2 + 3313/12288*a*x - 5005/4096*x^2 + 15409/61440*a + 11825/8192*x + 943/8192
a
0
0:
1
a
0
1:
1/2*a*x + 1/2*a + x
a
0
2:
1/8*a^2*x^2 + 1/4*a^2*x + 7/8*a*x^2 + 1/8*a^2 + 3/4*a*x + 3/2*x^2 - 1/8*a - 1/2
a
0
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 5/16*a^2*x^3 + 1/16*a^3*x + 9/16*a^2*x^2 + 37/24*a*x^3 + 1/48*a^3 + 3/16*a^2*x + 5/4*a*x^2 + 5/2*x^3 - 1/16*a^2 - 5/8*a*x - 1/3*a - 3/2*x
a
0
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 13/192*a^3*x^4 + 1/64*a^4*x^2 + 3/16*a^3*x^3 + 251/384*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 107/96*a^2*x^3 + 533/192*a*x^4 + 1/384*a^4 + 1/48*a^3*x + 11/64*a^2*x^2 + 35/16*a*x^3 + 35/8*x^4 - 1/64*a^3 - 37/96*a^2*x - 59/32*a*x^2 - 37/384*a^2 - 55/48*a*x - 15/4*x^2 + 7/64*a + 3/8
a
0
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 1/96*a^4*x^5 + 1/384*a^5*x^3 + 5/128*a^4*x^4 + 127/768*a^3*x^5 + 1/384*a^5*x^2 + 5/96*a^4*x^3 + 335/768*a^3*x^4 + 125/96*a^2*x^5 + 1/768*a^5*x + 5/192*a^4*x^2 + 115/384*a^3*x^3 + 275/128*a^2*x^4 + 1627/320*a*x^5 + 1/3840*a^5 - 25/384*a^3*x^2 - 5/96*a^2*x^3 + 63/16*a*x^4 + 63/8*x^5 - 1/384*a^4 - 85/768*a^3*x - 245/192*a^2*x^2 - 449/96*a*x^3 - 13/768*a^3 - 5/16*a^2*x - 49/16*a*x^2 - 35/4*x^3 + 25/384*a^2 + 47/64*a*x + 4/15*a + 15/8*x
a
1/2
0:
1
a
1/2
1:
1/2*a*x + 1/2*a + 5/4*x - 1/4
a
1/2
2:
1/8*a^2*x^2 + 1/4*a^2*x + a*x^2 + 1/8*a^2 + 3/4*a*x + 63/32*x^2 - 1/4*a - 7/16*x - 17/32
a
1/2
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 11/32*a^2*x^3 + 1/16*a^3*x + 19/32*a^2*x^2 + 359/192*a*x^3 + 1/48*a^3 + 5/32*a^2*x + 79/64*a*x^2 + 429/128*x^3 - 3/32*a^2 - 61/64*a*x - 99/128*x^2 - 61/192*a - 225/128*x + 23/128
a
1/2
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 7/96*a^3*x^4 + 1/64*a^4*x^2 + 19/96*a^3*x^3 + 583/768*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 29/24*a^2*x^3 + 1337/384*a*x^4 + 1/384*a^4 + 1/96*a^3*x + 7/128*a^2*x^2 + 821/384*a*x^3 + 12155/2048*x^4 - 1/48*a^3 - 23/48*a^2*x - 341/128*a*x^2 - 715/512*x^3 - 65/768*a^2 - 421/384*a*x - 4719/1024*x^2 + 43/192*a + 341/512*x + 827/2048
a
1/2
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 17/1536*a^4*x^5 + 1/384*a^5*x^3 + 21/512*a^4*x^4 + 287/1536*a^3*x^5 + 1/384*a^5*x^2 + 41/768*a^4*x^3 + 731/1536*a^3*x^4 + 4811/3072*a^2*x^5 + 1/768*a^5*x + 19/768*a^4*x^2 + 75/256*a^3*x^3 + 2423/1024*a^2*x^4 + 133443/20480*a*x^5 + 1/3840*a^5 - 1/512*a^4*x - 83/768*a^3*x^2 - 607/1536*a^2*x^3 + 46921/12288*a*x^4 + 88179/8192*x^5 - 5/1536*a^4 - 193/1536*a^3*x - 2345/1536*a^2*x^2 - 40771/6144*a*x^3 - 20995/8192*x^4 - 7/512*a^3 - 239/1024*a^2*x - 5993/2048*a*x^2 - 45305/4096*x^3 + 301/3072*a^2 + 15505/12288*a*x + 7605/4096*x^2 + 15329/61440*a + 17615/8192*x - 1207/8192
a
1
0:
1
a
1
1:
1/2*a*x + 1/2*a + 3/2*x - 1/2
a
1
2:
1/8*a^2*x^2 + 1/4*a^2*x + 9/8*a*x^2 + 1/8*a^2 + 3/4*a*x + 5/2*x^2 - 3/8*a - x - 1/2
a
1
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 3/8*a^2*x^3 + 1/16*a^3*x + 5/8*a^2*x^2 + 107/48*a*x^3 + 1/48*a^3 + 1/8*a^2*x + 19/16*a*x^2 + 35/8*x^3 - 1/8*a^2 - 21/16*a*x - 15/8*x^2 - 13/48*a - 15/8*x + 3/8
a
1
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 5/64*a^3*x^4 + 1/64*a^4*x^2 + 5/24*a^3*x^3 + 335/384*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 125/96*a^2*x^3 + 275/64*a*x^4 + 1/384*a^4 - 5/64*a^2*x^2 + 95/48*a*x^3 + 63/8*x^4 - 5/192*a^3 - 55/96*a^2*x - 115/32*a*x^2 - 7/2*x^3 - 25/384*a^2 - 15/16*a*x - 21/4*x^2 + 65/192*a + 3/2*x + 3/8
a
1
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 3/256*a^4*x^5 + 1/384*a^5*x^3 + 11/256*a^4*x^4 + 161/768*a^3*x^5 + 1/384*a^5*x^2 + 7/128*a^4*x^3 + 397/768*a^3*x^4 + 477/256*a^2*x^5 + 1/768*a^5*x + 3/128*a^4*x^2 + 109/384*a^3*x^3 + 661/256*a^2*x^4 + 15797/1920*a*x^5 + 1/3840*a^5 - 1/256*a^4*x - 59/384*a^3*x^2 - 103/128*a^2*x^3 + 1313/384*a*x^4 + 231/16*x^5 - 1/256*a^4 - 107/768*a^3*x - 227/128*a^2*x^2 - 1711/192*a*x^3 - 105/16*x^4 - 7/768*a^3 - 31/256*a^2*x - 475/192*a*x^2 - 105/8*x^3 + 33/256*a^2 + 701/384*a*x + 35/8*x^2 + 377/1920*a + 35/16*x - 5/16
a
3/2
0:
1
a
3/2
1:
1/2*a*x + 1/2*a + 7/4*x - 3/4
a
3/2
2:
1/8*a^2*x^2 + 1/4*a^2*x + 5/4*a*x^2 + 1/8*a^2 + 3/4*a*x + 99/32*x^2 - 1/2*a - 27/16*x - 13/32
a
3/2
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 13/32*a^2*x^3 + 1/16*a^3*x + 21/32*a^2*x^2 + 503/192*a*x^3 + 1/48*a^3 + 3/32*a^2*x + 71/64*a*x^2 + 715/128*x^3 - 5/32*a^2 - 109/64*a*x - 429/128*x^2 - 37/192*a - 231/128*x + 73/128
a
3/2
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 1/12*a^3*x^4 + 1/64*a^4*x^2 + 7/32*a^3*x^3 + 763/768*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 67/48*a^2*x^3 + 251/48*a*x^4 + 1/384*a^4 - 1/96*a^3*x - 29/128*a^2*x^2 + 217/128*a*x^3 + 20995/2048*x^4 - 1/32*a^3 - 2/3*a^2*x - 593/128*a*x^2 - 3315/512*x^3 - 29/768*a^2 - 251/384*a*x - 5655/1024*x^2 + 57/128*a + 1261/512*x + 579/2048
a
3/2
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 19/1536*a^4*x^5 + 1/384*a^5*x^3 + 23/512*a^4*x^4 + 359/1536*a^3*x^5 + 1/384*a^5*x^2 + 43/768*a^4*x^3 + 859/1536*a^3*x^4 + 6745/3072*a^2*x^5 + 1/768*a^5*x + 17/768*a^4*x^2 + 209/768*a^3*x^3 + 2861/1024*a^2*x^4 + 210003/20480*a*x^5 + 1/3840*a^5 - 3/512*a^4*x - 155/768*a^3*x^2 - 1973/1536*a^2*x^3 + 10947/4096*a*x^4 + 156009/8192*x^5 - 7/1536*a^4 - 233/1536*a^3*x - 3091/1536*a^2*x^2 - 70691/6144*a*x^3 - 101745/8192*x^4 - 5/1536*a^3 + 27/1024*a^2*x - 3353/2048*a*x^2 - 59755/4096*x^3 + 479/3072*a^2 + 9803/4096*a*x + 30855/4096*x^2 + 6529/61440*a + 15645/8192*x - 3917/8192
a
2
0:
1
a
2
1:
1/2*a*x + 1/2*a + 2*x - 1
a
2
2:
1/8*a^2*x^2 + 1/4*a^2*x + 11/8*a*x^2 + 1/8*a^2 + 3/4*a*x + 15/4*x^2 - 5/8*a - 5/2*x - 1/4
a
2
3:
1/48*a^3*x^3 + 1/16*a^3*x^2 + 7/16*a^2*x^3 + 1/16*a^3*x + 11/16*a^2*x^2 + 73/24*a*x^3 + 1/48*a^3 + 1/16*a^2*x + a*x^2 + 7*x^3 - 3/16*a^2 - 17/8*a*x - 21/4*x^2 - 1/12*a - 3/2*x + 3/4
a
2
4:
1/384*a^4*x^4 + 1/96*a^4*x^3 + 17/192*a^3*x^4 + 1/64*a^4*x^2 + 11/48*a^3*x^3 + 431/384*a^2*x^4 + 1/96*a^4*x + 5/32*a^3*x^2 + 143/96*a^2*x^3 + 1207/192*a*x^4 + 1/384*a^4 - 1/48*a^3*x - 25/64*a^2*x^2 + 61/48*a*x^3 + 105/8*x^4 - 7/192*a^3 - 73/96*a^2*x - 185/32*a*x^2 - 21/2*x^3 - 1/384*a^2 - 11/48*a*x - 21/4*x^2 + 103/192*a + 7/2*x + 1/8
a
2
5:
1/3840*a^5*x^5 + 1/768*a^5*x^4 + 5/384*a^4*x^5 + 1/384*a^5*x^3 + 3/64*a^4*x^4 + 199/768*a^3*x^5 + 1/384*a^5*x^2 + 11/192*a^4*x^3 + 463/768*a^3*x^4 + 985/384*a^2*x^5 + 1/768*a^5*x + 1/48*a^4*x^2 + 33/128*a^3*x^3 + 3*a^2*x^4 + 2021/160*a*x^5 + 1/3840*a^5 - 1/128*a^4*x - 97/384*a^3*x^2 - 353/192*a^2*x^3 + 289/192*a*x^4 + 99/4*x^5 - 1/192*a^4 - 125/768*a^3*x - 215/96*a^2*x^2 - 173/12*a*x^3 - 165/8*x^4 + 1/256*a^3 + 27/128*a^2*x - 11/32*a*x^2 - 15*x^3 + 17/96*a^2 + 281/96*a*x + 45/4*x^2 - 19/960*a + 5/4*x - 5/8
-1/2
0
0:
1
-1/2
0
1:
3/4*x - 1/4
-1/2
0
2:
35/32*x^2 - 5/16*x - 13/32
-1/2
0
3:
231/128*x^3 - 63/128*x^2 - 147/128*x + 19/128
-1/2
0
4:
6435/2048*x^4 - 429/512*x^3 - 2871/1024*x^2 + 243/512*x + 611/2048
-1/2
0
5:
46189/8192*x^5 - 12155/8192*x^4 - 26455/4096*x^3 + 5005/4096*x^2 + 11825/8192*x - 943/8192
-1/2
0
6:
676039/65536*x^6 - 88179/32768*x^5 - 944775/65536*x^4 + 47515/16384*x^3 + 325065/65536*x^2 - 19539/32768*x - 16169/65536
-1/2
0
7:
5014575/262144*x^7 - 1300075/262144*x^6 - 8268477/262144*x^5 + 1729665/262144*x^4 + 3855005/262144*x^3 - 558705/262144*x^2 - 442575/262144*x + 25499/262144
-1/2
0
8:
300540195/8388608*x^8 - 9694845/1048576*x^7 - 142748235/2097152*x^6 + 15340885/1048576*x^5 + 168229705/4194304*x^4 - 6798827/1048576*x^3 - 15698123/2097152*x^2 + 731459/1048576*x + 1803811/8388608
-1/2
0
9:
2268783825/33554432*x^9 - 583401555/33554432*x^8 - 1219839615/8388608*x^7 + 267463665/8388608*x^6 + 1738995615/16777216*x^5 - 302152725/16777216*x^4 - 227837379/8388608*x^3 + 26766117/8388608*x^2 + 63821817/33554432*x - 2877659/33554432
-1/2
0
10:
34461632205/268435456*x^10 - 4418157975/134217728*x^9 - 82750904775/268435456*x^8 + 2302900875/33554432*x^7 + 34617950325/134217728*x^6 - 3166072245/67108864*x^5 - 11755266075/134217728*x^4 + 397231275/33554432*x^3 + 2773596105/268435456*x^2 - 105510615/134217728*x - 51825539/268435456
-1/2
1/2
0:
1
-1/2
1/2
1:
x - 1/2
-1/2
1/2
2:
3/2*x^2 - 3/4*x - 3/8
-1/2
1/2
3:
5/2*x^3 - 5/4*x^2 - 5/4*x + 5/16
-1/2
1/2
4:
35/128*(8*x^3 - 6*x - 1)*(2*x - 1)
-1/2
1/2
5:
63/8*x^5 - 63/16*x^4 - 63/8*x^3 + 189/64*x^2 + 189/128*x - 63/256
-1/2
1/2
6:
231/16*x^6 - 231/32*x^5 - 1155/64*x^4 + 231/32*x^3 + 693/128*x^2 - 693/512*x - 231/1024
-1/2
1/2
7:
429/2048*(16*x^4 + 8*x^3 - 16*x^2 - 8*x + 1)*(4*x^2 - 2*x - 1)*(2*x - 1)
-1/2
1/2
8:
6435/128*x^8 - 6435/256*x^7 - 45045/512*x^6 + 19305/512*x^5 + 96525/2048*x^4 - 32175/2048*x^3 - 32175/4096*x^2 + 6435/4096*x + 6435/32768
-1/2
1/2
9:
12155/128*x^9 - 12155/256*x^8 - 12155/64*x^7 + 85085/1024*x^6 + 255255/2048*x^5 - 182325/4096*x^4 - 60775/2048*x^3 + 60775/8192*x^2 + 60775/32768*x - 12155/65536
-1/2
1/2
10:
46189/262144*(64*x^6 + 32*x^5 - 96*x^4 - 48*x^3 + 32*x^2 + 16*x + 1)*(8*x^3 - 4*x^2 - 4*x + 1)*(2*x - 1)
-1/2
1
0:
1
-1/2
1
1:
5/4*x - 3/4
-1/2
1
2:
63/32*x^2 - 21/16*x - 9/32
-1/2
1
3:
429/128*x^3 - 297/128*x^2 - 153/128*x + 61/128
-1/2
1
4:
12155/2048*x^4 - 2145/512*x^3 - 3575/1024*x^2 + 935/512*x + 395/2048
-1/2
1
5:
88179/8192*x^5 - 62985/8192*x^4 - 36465/4096*x^3 + 21255/4096*x^2 + 10335/8192*x - 3093/8192
-1/2
1
6:
1300075/65536*x^6 - 468027/32768*x^5 - 1390515/65536*x^4 + 214795/16384*x^3 + 337365/65536*x^2 - 72555/32768*x - 10157/65536
-1/2
1
7:
9694845/262144*x^7 - 7020405/262144*x^6 - 12740735/262144*x^5 + 8164471/262144*x^4 + 4474519/262144*x^3 - 2249695/262144*x^2 - 352597/262144*x + 84509/262144
-1/2
1
8:
583401555/8388608*x^8 - 53036505/1048576*x^7 - 227543715/2097152*x^6 + 74746665/1048576*x^5 + 211300425/4194304*x^4 - 28990143/1048576*x^3 - 14592627/2097152*x^2 + 2664351/1048576*x + 1115379/8388608
-1/2
1
9:
4418157975/33554432*x^9 - 3224061225/33554432*x^8 - 1995847425/8388608*x^7 + 1335217275/8388608*x^6 + 2316407625/16777216*x^5 - 1344635775/16777216*x^4 - 236174925/8388608*x^3 + 104349735/8388608*x^2 + 48000015/33554432*x - 9595745/33554432
-1/2
1
10:
67282234305/268435456*x^10 - 24615451575/134217728*x^9 - 138225228075/268435456*x^8 + 11719206675/33554432*x^7 + 48258567225/134217728*x^6 - 14564568645/67108864*x^5 - 13218155775/134217728*x^4 + 1630275075/33554432*x^3 + 2393942925/268435456*x^2 - 379711255/134217728*x - 31728279/268435456
-1/2
3/2
0:
1
-1/2
3/2
1:
3/2*x - 1
-1/2
3/2
2:
5/2*x^2 - 2*x - 1/8
-1/2
3/2
3:
35/8*x^3 - 15/4*x^2 - 15/16*x + 5/8
-1/2
3/2
4:
63/8*x^4 - 7*x^3 - 105/32*x^2 + 21/8*x + 7/128
-1/2
3/2
5:
231/16*x^5 - 105/8*x^4 - 147/16*x^3 + 63/8*x^2 + 189/256*x - 63/128
-1/2
3/2
6:
429/16*x^6 - 99/4*x^5 - 1485/64*x^4 + 165/8*x^3 + 495/128*x^2 - 99/32*x - 33/1024
-1/2
3/2
7:
6435/128*x^7 - 3003/64*x^6 - 14157/256*x^5 + 6435/128*x^4 + 15015/1024*x^3 - 6435/512*x^2 - 1287/2048*x + 429/1024
-1/2
3/2
8:
12155/128*x^8 - 715/8*x^7 - 65065/512*x^6 + 15015/128*x^5 + 96525/2048*x^4 - 10725/256*x^3 - 17875/4096*x^2 + 3575/1024*x + 715/32768
-1/2
3/2
9:
46189/256*x^9 - 21879/128*x^8 - 36465/128*x^7 + 17017/64*x^6 + 561561/4096*x^5 - 255255/2048*x^4 - 85085/4096*x^3 + 36465/2048*x^2 + 36465/65536*x - 12155/32768
-1/2
3/2
10:
88179/256*x^10 - 20995/64*x^9 - 642447/1024*x^8 + 37791/64*x^7 + 382109/1024*x^6 - 88179/256*x^5 - 1322685/16384*x^4 + 146965/2048*x^3 + 314925/65536*x^2 - 62985/16384*x - 4199/262144
-1/2
2
0:
1
-1/2
2
1:
7/4*x - 5/4
-1/2
2
2:
99/32*x^2 - 45/16*x + 3/32
-1/2
2
3:
715/128*x^3 - 715/128*x^2 - 55/128*x + 95/128
-1/2
2
4:
20995/2048*x^4 - 5525/512*x^3 - 2535/1024*x^2 + 1755/512*x - 285/2048
-1/2
2
5:
156009/8192*x^5 - 169575/8192*x^4 - 33915/4096*x^3 + 44625/4096*x^2 - 1155/8192*x - 4683/8192
-1/2
2
6:
2340135/65536*x^6 - 1300075/32768*x^5 - 1508087/65536*x^4 + 486115/16384*x^3 + 83657/65536*x^2 - 127211/32768*x + 9191/65536
-1/2
2
7:
17678835/262144*x^7 - 19959975/262144*x^6 - 15279705/262144*x^5 + 19501125/262144*x^4 + 2211657/262144*x^3 - 4358277/262144*x^2 + 135261/262144*x + 125991/262144
-1/2
2
8:
1074687075/8388608*x^8 - 153526725/1048576*x^7 - 293096475/2097152*x^6 + 185942925/1048576*x^5 + 142907625/4194304*x^4 - 60387075/1048576*x^3 + 644805/2097152*x^2 + 4480875/1048576*x - 1128285/8388608
-1/2
2
9:
8205150525/33554432*x^9 - 9467481375/33554432*x^8 - 2712305475/8388608*x^7 + 3428763525/8388608*x^6 + 1893496275/16777216*x^5 - 2959929225/16777216*x^4 - 47532375/8388608*x^3 + 191704425/8388608*x^2 - 26933115/33554432*x - 14167175/33554432
-1/2
2
10:
125788525005/268435456*x^10 - 73132863375/134217728*x^9 - 195853375575/268435456*x^8 + 30872221875/33554432*x^7 + 44903229525/134217728*x^6 - 33486181245/67108864*x^5 - 5150932875/134217728*x^4 + 3204969075/33554432*x^3 - 592032375/268435456*x^2 - 620898575/134217728*x + 34308461/268435456
0
1/2
0:
1
0
1/2
1:
5/4*x - 1/4
0
1/2
2:
63/32*x^2 - 7/16*x - 17/32
0
1/2
3:
429/128*x^3 - 99/128*x^2 - 225/128*x + 23/128
0
1/2
4:
12155/2048*x^4 - 715/512*x^3 - 4719/1024*x^2 + 341/512*x + 827/2048
0
1/2
5:
88179/8192*x^5 - 20995/8192*x^4 - 45305/4096*x^3 + 7605/4096*x^2 + 17615/8192*x - 1207/8192
0
1/2
6:
1300075/65536*x^6 - 156009/32768*x^5 - 1661835/65536*x^4 + 75905/16384*x^3 + 512805/65536*x^2 - 27465/32768*x - 22181/65536
0
1/2
7:
9694845/262144*x^7 - 2340135/262144*x^6 - 14820855/262144*x^5 + 2860165/262144*x^4 + 6319495/262144*x^3 - 834309/262144*x^2 - 650573/262144*x + 33511/262144
0
1/2
8:
583401555/8388608*x^8 - 17678835/1048576*x^7 - 259479675/2097152*x^6 + 26016795/1048576*x^5 + 283492825/4194304*x^4 - 10593317/1048576*x^3 - 24190971/2097152*x^2 + 1029325/1048576*x + 2492243/8388608
0
1/2
9:
4418157975/33554432*x^9 - 1074687075/33554432*x^8 - 2241490185/8388608*x^7 + 462441105/8388608*x^6 + 2992105305/16777216*x^5 - 485885925/16777216*x^4 - 363397125/8388608*x^3 + 39547557/8388608*x^2 + 93079791/33554432*x - 3840427/33554432
0
1/2
10:
67282234305/268435456*x^10 - 8205150525/134217728*x^9 - 153373198275/268435456*x^8 + 4042870425/33554432*x^7 + 60540705225/134217728*x^6 - 5217117255/67108864*x^5 - 19248769575/134217728*x^4 + 608823225/33554432*x^3 + 4209857325/268435456*x^2 - 148592765/134217728*x - 71922799/268435456
0
1
0:
1
0
1
1:
3/2*x - 1/2
0
1
2:
5/2*x^2 - x - 1/2
0
1
3:
35/8*x^3 - 15/8*x^2 - 15/8*x + 3/8
0
1
4:
63/8*x^4 - 7/2*x^3 - 21/4*x^2 + 3/2*x + 3/8
0
1
5:
231/16*x^5 - 105/16*x^4 - 105/8*x^3 + 35/8*x^2 + 35/16*x - 5/16
0
1
6:
429/16*x^6 - 99/8*x^5 - 495/16*x^4 + 45/4*x^3 + 135/16*x^2 - 15/8*x - 5/16
0
1
7:
6435/128*x^7 - 3003/128*x^6 - 9009/128*x^5 + 3465/128*x^4 + 3465/128*x^3 - 945/128*x^2 - 315/128*x + 35/128
0
1
8:
12155/128*x^8 - 715/16*x^7 - 5005/32*x^6 + 1001/16*x^5 + 5005/64*x^4 - 385/16*x^3 - 385/32*x^2 + 35/16*x + 35/128
0
1
9:
46189/256*x^9 - 21879/256*x^8 - 21879/64*x^7 + 9009/64*x^6 + 27027/128*x^5 - 9009/128*x^4 - 3003/64*x^3 + 693/64*x^2 + 693/256*x - 63/256
0
1
10:
88179/256*x^10 - 20995/128*x^9 - 188955/256*x^8 + 9945/32*x^7 + 69615/128*x^6 - 12285/64*x^5 - 20475/128*x^4 + 1365/32*x^3 + 4095/256*x^2 - 315/128*x - 63/256
0
3/2
0:
1
0
3/2
1:
7/4*x - 3/4
0
3/2
2:
99/32*x^2 - 27/16*x - 13/32
0
3/2
3:
715/128*x^3 - 429/128*x^2 - 231/128*x + 73/128
0
3/2
4:
20995/2048*x^4 - 3315/512*x^3 - 5655/1024*x^2 + 1261/512*x + 579/2048
0
3/2
5:
156009/8192*x^5 - 101745/8192*x^4 - 59755/4096*x^3 + 30855/4096*x^2 + 15645/8192*x - 3917/8192
0
3/2
6:
2340135/65536*x^6 - 780045/32768*x^5 - 2340135/65536*x^4 + 327845/16384*x^3 + 528105/65536*x^2 - 99501/32768*x - 14857/65536
0
3/2
7:
17678835/262144*x^7 - 11975985/262144*x^6 - 21887145/262144*x^5 + 12924275/262144*x^4 + 7203945/262144*x^3 - 3261027/262144*x^2 - 530803/262144*x + 110049/262144
0
3/2
8:
1074687075/8388608*x^8 - 92116035/1048576*x^7 - 397308555/2097152*x^6 + 121650795/1048576*x^5 + 348086025/4194304*x^4 - 43779925/1048576*x^3 - 22593291/2097152*x^2 + 3684093/1048576*x + 1625443/8388608
0
3/2
9:
8205150525/33554432*x^9 - 5680488825/33554432*x^8 - 3531114675/8388608*x^7 + 2221019955/8388608*x^6 + 3888413235/16777216*x^5 - 2095797375/16777216*x^4 - 374675175/8388608*x^3 + 150908175/8388608*x^2 + 71813061/33554432*x - 12705169/33554432
0
3/2
10:
125788525005/268435456*x^10 - 43879718025/134217728*x^9 - 247224752775/268435456*x^8 + 19840547925/33554432*x^7 + 82283649525/134217728*x^6 - 23278656555/67108864*x^5 - 21407370075/134217728*x^4 + 2441452725/33554432*x^3 + 3671418825/268435456*x^2 - 527588425/134217728*x - 46080579/268435456
0
2
0:
1
0
2
1:
2*x - 1
0
2
2:
15/4*x^2 - 5/2*x - 1/4
0
2
3:
7*x^3 - 21/4*x^2 - 3/2*x + 3/4
0
2
4:
105/8*x^4 - 21/2*x^3 - 21/4*x^2 + 7/2*x + 1/8
0
2
5:
99/4*x^5 - 165/8*x^4 - 15*x^3 + 45/4*x^2 + 5/4*x - 5/8
0
2
6:
3003/64*x^6 - 1287/32*x^5 - 2475/64*x^4 + 495/16*x^3 + 405/64*x^2 - 135/32*x - 5/64
0
2
7:
715/8*x^7 - 5005/64*x^6 - 3003/32*x^5 + 5005/64*x^4 + 385/16*x^3 - 1155/64*x^2 - 35/32*x + 35/64
0
2
8:
21879/128*x^8 - 2431/16*x^7 - 7007/32*x^6 + 3003/16*x^5 + 5005/64*x^4 - 1001/16*x^3 - 231/32*x^2 + 77/16*x + 7/128
0
2
9:
20995/64*x^9 - 37791/128*x^8 - 1989/4*x^7 + 13923/32*x^6 + 7371/32*x^5 - 12285/64*x^4 - 273/8*x^3 + 819/32*x^2 + 63/64*x - 63/128
0
2
10:
323323/512*x^10 - 146965/256*x^9 - 566865/512*x^8 + 62985/64*x^7 + 162435/256*x^6 - 69615/128*x^5 - 34125/256*x^4 + 6825/64*x^3 + 4095/512*x^2 - 1365/256*x - 21/512
1/2
1
0:
1
1/2
1
1:
7/4*x - 1/4
1/2
1
2:
99/32*x^2 - 9/16*x - 21/32
1/2
1
3:
715/128*x^3 - 143/128*x^2 - 319/128*x + 27/128
1/2
1
4:
20995/2048*x^4 - 1105/512*x^3 - 7215/1024*x^2 + 455/512*x + 1075/2048
1/2
1
5:
156009/8192*x^5 - 33915/8192*x^4 - 72675/4096*x^3 + 10965/4096*x^2 + 25005/8192*x - 1503/8192
1/2
1
6:
2340135/65536*x^6 - 260015/32768*x^5 - 2756159/65536*x^4 + 115311/16384*x^3 + 771001/65536*x^2 - 37247/32768*x - 29505/65536
1/2
1
7:
17678835/262144*x^7 - 3991995/262144*x^6 - 25190865/262144*x^5 + 4512025/262144*x^4 + 9895865/262144*x^3 - 1199793/262144*x^2 - 923419/262144*x + 43027/262144
1/2
1
8:
1074687075/8388608*x^8 - 30705345/1048576*x^7 - 449414595/2097152*x^6 + 42231105/1048576*x^5 + 457630425/4194304*x^4 - 15902775/1048576*x^3 - 35974323/2097152*x^2 + 1408743/1048576*x + 3359043/8388608
1/2
1
9:
8205150525/33554432*x^9 - 1893496275/33554432*x^8 - 3940519275/8388608*x^7 + 767633625/8388608*x^6 + 4945421475/16777216*x^5 - 753826725/16777216*x^4 - 559774575/8388608*x^3 + 56747325/8388608*x^2 + 132076005/33554432*x - 5024315/33554432
1/2
1
10:
125788525005/268435456*x^10 - 14626572675/134217728*x^9 - 272910441375/268435456*x^8 + 6833051775/33554432*x^7 + 101970670725/134217728*x^6 - 8307798345/67108864*x^5 - 30480617475/134217728*x^4 + 906159375/33554432*x^3 + 6212594025/268435456*x^2 - 204560675/134217728*x - 97765019/268435456
1/2
3/2
0:
1
1/2
3/2
1:
2*x - 1/2
1/2
3/2
2:
15/4*x^2 - 5/4*x - 5/8
1/2
3/2
3:
7*x^3 - 21/8*x^2 - 21/8*x + 7/16
1/2
3/2
4:
105/8*x^4 - 21/4*x^3 - 63/8*x^2 + 63/32*x + 63/128
1/2
3/2
5:
99/4*x^5 - 165/16*x^4 - 165/8*x^3 + 99/16*x^2 + 99/32*x - 99/256
1/2
3/2
6:
3003/64*x^6 - 1287/64*x^5 - 6435/128*x^4 + 2145/128*x^3 + 6435/512*x^2 - 1287/512*x - 429/1024
1/2
3/2
7:
715/8*x^7 - 5005/128*x^6 - 15015/128*x^5 + 10725/256*x^4 + 10725/256*x^3 - 10725/1024*x^2 - 3575/1024*x + 715/2048
1/2
3/2
8:
21879/128*x^8 - 2431/32*x^7 - 17017/64*x^6 + 51051/512*x^5 + 255255/2048*x^4 - 36465/1024*x^3 - 36465/2048*x^2 + 12155/4096*x + 12155/32768
1/2
3/2
9:
20995/64*x^9 - 37791/256*x^8 - 37791/64*x^7 + 29393/128*x^6 + 88179/256*x^5 - 440895/4096*x^4 - 146965/2048*x^3 + 62985/4096*x^2 + 62985/16384*x - 20995/65536
1/2
3/2
10:
323323/512*x^10 - 146965/512*x^9 - 1322685/1024*x^8 + 264537/512*x^7 + 1851759/2048*x^6 - 617253/2048*x^5 - 1028755/4096*x^4 + 1028755/16384*x^3 + 3086265/131072*x^2 - 440895/131072*x - 88179/262144
1/2
2
0:
1
1/2
2
1:
9/4*x - 3/4
1/2
2
2:
143/32*x^2 - 33/16*x - 17/32
1/2
2
3:
1105/128*x^3 - 585/128*x^2 - 325/128*x + 85/128
1/2
2
4:
33915/2048*x^4 - 4845/512*x^3 - 8415/1024*x^2 + 1635/512*x + 795/2048
1/2
2
5:
260015/8192*x^5 - 156009/8192*x^4 - 92701/4096*x^3 + 42959/4096*x^2 + 22491/8192*x - 4837/8192
1/2
2
6:
3991995/65536*x^6 - 1238895/32768*x^5 - 3747275/65536*x^4 + 480263/16384*x^3 + 789621/65536*x^2 - 132335/32768*x - 20741/65536
1/2
2
7:
30705345/262144*x^7 - 19539765/262144*x^6 - 35927955/262144*x^5 + 19670175/262144*x^4 + 11121075/262144*x^3 - 4577391/262144*x^2 - 768033/262144*x + 140229/262144
1/2
2
8:
1893496275/8388608*x^8 - 153526725/1048576*x^7 - 665282475/2097152*x^6 + 190745325/1048576*x^5 + 551525625/4194304*x^4 - 64030275/1048576*x^3 - 33739275/2097152*x^2 + 4969035/1048576*x + 2285715/8388608
1/2
2
9:
14626572675/33554432*x^9 - 9632133225/33554432*x^8 - 6009792525/8388608*x^7 + 3566715075/8388608*x^6 + 6294595725/16777216*x^5 - 3166210575/16777216*x^4 - 574702425/8388608*x^3 + 212644575/8388608*x^2 + 104084475/33554432*x - 16509185/33554432
1/2
2
10:
226419345009/268435456*x^10 - 75473115003/134217728*x^9 - 426510859203/268435456*x^8 + 32492396079/33554432*x^7 + 135610556697/134217728*x^6 - 36103878657/67108864*x^5 - 33588176391/134217728*x^4 + 3562087023/33554432*x^3 + 5466783069/268435456*x^2 - 717933051/134217728*x - 65035311/268435456
1
3/2
0:
1
1
3/2
1:
9/4*x - 1/4
1
3/2
2:
143/32*x^2 - 11/16*x - 25/32
1
3/2
3:
1105/128*x^3 - 195/128*x^2 - 429/128*x + 31/128
1
3/2
4:
33915/2048*x^4 - 1615/512*x^3 - 10455/1024*x^2 + 585/512*x + 1355/2048
1
3/2
5:
260015/8192*x^5 - 52003/8192*x^4 - 110789/4096*x^3 + 15181/4096*x^2 + 34187/8192*x - 1831/8192
1
3/2
6:
3991995/65536*x^6 - 412965/32768*x^5 - 4359075/65536*x^4 + 168245/16384*x^3 + 1115205/65536*x^2 - 49077/32768*x - 38269/65536
1
3/2
7:
30705345/262144*x^7 - 6513255/262144*x^6 - 40970475/262144*x^5 + 6846525/262144*x^4 + 14920675/262144*x^3 - 1672629/262144*x^2 - 1273209/262144*x + 54175/262144
1
3/2
8:
1893496275/8388608*x^8 - 51175575/1048576*x^7 - 747163395/2097152*x^6 + 66063015/1048576*x^5 + 712406025/4194304*x^4 - 23137425/1048576*x^3 - 51913875/2097152*x^2 + 1883217/1048576*x + 4432371/8388608
1
3/2
9:
14626572675/33554432*x^9 - 3210711075/33554432*x^8 - 6668399925/8388608*x^7 + 1230438825/8388608*x^6 + 7896613725/16777216*x^5 - 1134155925/16777216*x^4 - 836903025/8388608*x^3 + 79411725/8388608*x^2 + 183037275/33554432*x - 6460555/33554432
1
3/2
10:
226419345009/268435456*x^10 - 25157705001/134217728*x^9 - 468635388507/268435456*x^8 + 11173274541/33554432*x^7 + 166222628649/134217728*x^6 - 12842043291/67108864*x^5 - 46880653743/134217728*x^4 + 1314811773/33554432*x^3 + 8946435069/268435456*x^2 - 276081993/134217728*x - 130494727/268435456
1
2
0:
1
1
2
1:
5/2*x - 1/2
1
2
2:
21/4*x^2 - 3/2*x - 3/4
1
2
3:
21/2*x^3 - 7/2*x^2 - 7/2*x + 1/2
1
2
4:
165/8*x^4 - 15/2*x^3 - 45/4*x^2 + 5/2*x + 5/8
1
2
5:
1287/32*x^5 - 495/32*x^4 - 495/16*x^3 + 135/16*x^2 + 135/32*x - 15/32
1
2
6:
5005/64*x^6 - 1001/32*x^5 - 5005/64*x^4 + 385/16*x^3 + 1155/64*x^2 - 105/32*x - 35/64
1
2
7:
2431/16*x^7 - 1001/16*x^6 - 3003/16*x^5 + 1001/16*x^4 + 1001/16*x^3 - 231/16*x^2 - 77/16*x + 7/16
1
2
8:
37791/128*x^8 - 1989/16*x^7 - 13923/32*x^6 + 2457/16*x^5 + 12285/64*x^4 - 819/16*x^3 - 819/32*x^2 + 63/16*x + 63/128
1
2
9:
146965/256*x^9 - 62985/256*x^8 - 62985/64*x^7 + 23205/64*x^6 + 69615/128*x^5 - 20475/128*x^4 - 6825/64*x^3 + 1365/64*x^2 + 1365/256*x - 105/256
1
2
10:
572033/512*x^10 - 124355/256*x^9 - 1119195/512*x^8 + 53295/64*x^7 + 373065/256*x^6 - 58905/128*x^5 - 98175/256*x^4 + 5775/64*x^3 + 17325/512*x^2 - 1155/256*x - 231/512
3/2
2
0:
1
3/2
2
1:
11/4*x - 1/4
3/2
2
2:
195/32*x^2 - 13/16*x - 29/32
3/2
2
3:
1615/128*x^3 - 255/128*x^2 - 555/128*x + 35/128
3/2
2
4:
52003/2048*x^4 - 2261/512*x^3 - 14535/1024*x^2 + 731/512*x + 1667/2048
3/2
2
5:
412965/8192*x^5 - 76475/8192*x^4 - 162127/4096*x^3 + 20349/4096*x^2 + 45353/8192*x - 2191/8192
3/2
2
6:
6513255/65536*x^6 - 630315/32768*x^5 - 6629175/65536*x^4 + 237475/16384*x^3 + 1562505/65536*x^2 - 63147/32768*x - 48601/65536
3/2
2
7:
51175575/262144*x^7 - 10235115/262144*x^6 - 64202085/262144*x^5 + 10055025/262144*x^4 + 21791925/262144*x^3 - 2271825/262144*x^2 - 1713063/262144*x + 67083/262144
3/2
2
8:
3210711075/8388608*x^8 - 82325925/1048576*x^7 - 1199288475/2097152*x^6 + 100126125/1048576*x^5 + 1075091625/4194304*x^4 - 32775075/1048576*x^3 - 73014075/2097152*x^2 + 2467275/1048576*x + 5742435/8388608
3/2
2
9:
25157705001/33554432*x^9 - 5265566163/33554432*x^8 - 10916417655/8388608*x^7 + 1913254497/8388608*x^6 + 12236480487/16777216*x^5 - 1661559669/16777216*x^4 - 1219224699/8388608*x^3 + 108739125/8388608*x^2 + 248503761/33554432*x - 8182427/33554432
3/2
2
10:
394137378349/268435456*x^10 - 41929508335/134217728*x^9 - 779888855031/268435456*x^8 + 17746908179/33554432*x^7 + 263273551541/134217728*x^6 - 19340006301/67108864*x^5 - 70283842475/134217728*x^4 + 1865648851/33554432*x^3 + 12609276009/268435456*x^2 - 366166383/134217728*x - 171395763/268435456
Definition
The Jacobi polynomials $P_n^{(\alpha,\beta)}$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (3) such that $\deg P_n^{(\alpha,\beta)}=n$ and $P_n^{(\alpha,\beta)}(1) = \binom{n+\alpha}{n}$. Due to (4) and (2) we restrict to $\alpha < \beta$.
Parameters
$\alpha$
—   real number ($\alpha > -1$)
$\beta$
—   real number ($\beta > -1$)
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$\sum_{n=0}^\infty P_n^{(\alpha,\beta)}(x)t^n = \frac{2^{\alpha+\beta}}{R(1-t+R)^\alpha (1+t+R)^\beta}$, where $R = (1-2zt+t)^{1/2}$, defined via the standard branch of the square root (generating function).
(2)
$P_n^{(\alpha,\beta)}(-x) = (-1)^n P_n^{(\beta,\alpha)}(x)$ (symmetry).
(3)
The $P_n^{(\alpha,\beta)}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x)^{\alpha}(1+x)^{\beta} dx$.
(4)
The polynomials with $\alpha=\beta$ are called Gegenbauer polynomials (with parameter $\alpha+1/2$).
(5)
The polynomials with $\alpha=\beta=\pm 1/2$ are called Chebyshev polynomials of the second kind and Chebyshev polynomials of the first kind, respectively.
(6)
The polynomials with $\alpha=\beta=0$ are called Legendre polynomials.
Programs
(P1)
Sage
polynomials = {(alpha,beta):
{n: jacobi_P(n,alpha,beta,x) for n in [0..10]}
for alpha, beta in cartesian_product([[0..2],[0..2]])
}