add Shifted Gegenbauer polynomials

Author: whydenyscry

README.md

@@ -494,6 +494,190 @@ $$\begin{align*}
 
 ### Shifted Gegenbauer polynomials
 
+$$
+C_n^{(\lambda)\ast}(x) = C_n^{(\lambda)}(2x-1).
+$$
+
+$$\begin{align*}
+    C_{0}^{(1)\ast}(x)&=1,\\
+    C_{1}^{(1)\ast}(x)&=4x-2,\\
+    C_{2}^{(1)\ast}(x)&=16x^2-16x+3,\\
+    C_{3}^{(1)\ast}(x)&=64x^3-96x^2+40x-4,\\
+    C_{4}^{(1)\ast}(x)&=256x^4-512x^3+336x^2-80x+5,\\
+    C_{5}^{(1)\ast}(x)&=1024x^5-2560x^4+2304x^3-896x^2+140x-6,\\
+    C_{6}^{(1)\ast}(x)&=4096x^6-12288x^5+14080x^4-7680x^3+2016x^2-224x+7,\\
+    C_{7}^{(1)\ast}(x)&=16384x^7-57344x^6+79872x^5-56320x^4+21120x^3-4032x^2+336x-8,\\
+    C_{8}^{(1)\ast}(x)&=65536x^8-262144x^7+430080x^6-372736x^5+183040x^4-50688x^3+7392x^2-480x+9,\\
+    C_{9}^{(1)\ast}(x)&=262144x^9-1179648x^8+2228224x^7-2293760x^6+1397760x^5-512512x^4+109824x^3-12672x^2+660x-10,\\
+    C_{10}^{(1)\ast}(x)&=1048576x^{10}-5242880x^9+11206656x^8-13369344x^7+9748480x^6-4472832x^5+1281280x^4-219648x^3+20592x^2-880x+11.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-1.svg"/>
+</p>
+
+$$\begin{align*}
+    C_{0}^{(2)\ast}(x)&=1,\\
+    C_{1}^{(2)\ast}(x)&=8x-4,\\
+    C_{2}^{(2)\ast}(x)&=48x^2-48x+10,\\
+    C_{3}^{(2)\ast}(x)&=256x^3-384x^2+168x-20,\\
+    C_{4}^{(2)\ast}(x)&=1280x^4-2560x^3+1728x^2-448x+35,\\
+    C_{5}^{(2)\ast}(x)&=6144x^5-15360x^4+14080x^3-5760x^2+1008x-56,\\
+    C_{6}^{(2)\ast}(x)&=28672x^6-86016x^5+99840x^4-56320x^3+15840x^2-2016x+84,\\
+    C_{7}^{(2)\ast}(x)&=131072x^7-458752x^6+645120x^5-465920x^4+183040x^3-38016x^2+3696x-120,\\
+    C_{8}^{(2)\ast}(x)&=589824x^8-2359296x^7+3899392x^6-3440640x^5+1747200x^4-512512x^3+82368x^2-6336x+165,\\
+    C_{9}^{(2)\ast}(x)&=2621440x^9-11796480x^8+22413312x^7-23396352x^6+14622720x^5-5591040x^4+1281280x^3-164736x^2+10296x-220,\\
+    C_{10}^{(2)\ast}(x)&=11534336x^{10}-57671680x^9+123863040x^8-149422080x^7+111132672x^6-52641792x^5+15841280x^4-2928640x^3+308880x^2-16016x+286.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-2.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(3)\ast}(x)&=1,\\
+    C_{1}^{(3)\ast}(x)&=12x-6,\\
+    C_{2}^{(3)\ast}(x)&=96x^2-96x+21,\\
+    C_{3}^{(3)\ast}(x)&=640x^3-960x^2+432x-56,\\
+    C_{4}^{(3)\ast}(x)&=3840x^4-7680x^3+5280x^2-1440x+126,\\
+    C_{5}^{(3)\ast}(x)&=21504x^5-53760x^4+49920x^3-21120x^2+3960x-252,\\
+    C_{6}^{(3)\ast}(x)&=114688x^6-344064x^5+403200x^4-232960x^3+68640x^2-9504x+462,\\
+    C_{7}^{(3)\ast}(x)&=589824x^7-2064384x^6+2924544x^5-2150400x^4+873600x^3-192192x^2+20592x-792,\\
+    C_{8}^{(3)\ast}(x)&=2949120x^8-11796480x^7+19611648x^6-17547264x^5+9139200x^4-2795520x^3+480480x^2-41184x+1287,\\
+    C_{9}^{(3)\ast}(x)&=14417920x^9-64880640x^8+123863040x^7-130744320x^6+83349504x^5-32901120x^4+7920640x^3-1098240x^2+77220x-2002,\\
+    C_{10}^{(3)\ast}(x)&=69206016x^{10}-346030080x^9+746127360x^8-908328960x^7+686407680x^6-333398016x^5+104186880x^4-20367360x^3+2333760x^2-137280x+3003.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-3.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(4)\ast}(x)&=1,\\
+    C_{1}^{(4)\ast}(x)&=16x-8,\\
+    C_{2}^{(4)\ast}(x)&=160x^2-160x+36,\\
+    C_{3}^{(4)\ast}(x)&=1280x^3-1920x^2+880x-120,\\
+    C_{4}^{(4)\ast}(x)&=8960x^4-17920x^3+12480x^2-3520x+330,\\
+    C_{5}^{(4)\ast}(x)&=57344x^5-143360x^4+134400x^3-58240x^2+11440x-792,\\
+    C_{6}^{(4)\ast}(x)&=344064x^6-1032192x^5+1218560x^4-716800x^3+218400x^2-32032x+1716,\\
+    C_{7}^{(4)\ast}(x)&=1966080x^7-6881280x^6+9805824x^5-7311360x^4+3046400x^3-698880x^2+80080x-3432,\\
+    C_{8}^{(4)\ast}(x)&=10813440x^8-43253760x^7+72253440x^6-65372160x^5+34728960x^4-10967040x^3+1980160x^2-183040x+6435,\\
+    C_{9}^{(4)\ast}(x)&=57671680x^9-259522560x^8+497418240x^7-529858560x^6+343203840x^5-138915840x^4+34728960x^3-5091840x^2+388960x-11440,\\
+    C_{10}^{(4)\ast}(x)&=299892736x^{10}-1499463680x^9+3244032000x^8-3979345920x^7+3046686720x^6-1510096896x^5+486205440x^4-99225600x^3+12093120x^2-777920x+19448.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-4.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(5)\ast}(x)&=1,\\
+    C_{1}^{(5)\ast}(x)&=20x-10,\\
+    C_{2}^{(5)\ast}(x)&=240x^2-240x+55,\\
+    C_{3}^{(5)\ast}(x)&=2240x^3-3360x^2+1560x-220,\\
+    C_{4}^{(5)\ast}(x)&=17920x^4-35840x^3+25200x^2-7280x+715,\\
+    C_{5}^{(5)\ast}(x)&=129024x^5-322560x^4+304640x^3-134400x^2+27300x-2002,\\
+    C_{6}^{(5)\ast}(x)&=860160x^6-2580480x^5+3064320x^4-1827840x^3+571200x^2-87360x+5005,\\
+    C_{7}^{(5)\ast}(x)&=5406720x^7-18923520x^6+27095040x^5-20428800x^4+8682240x^3-2056320x^2+247520x-11440,\\
+    C_{8}^{(5)\ast}(x)&=32440320x^8-129761280x^7+217620480x^6-198696960x^5+107251200x^4-34728960x^3+6511680x^2-636480x+24310,\\
+    C_{9}^{(5)\ast}(x)&=187432960x^9-843448320x^8+1622016000x^7-1740963840x^6+1142507520x^5-471905280x^4+121551360x^3-18604800x^2+1511640x-48620,\\
+    C_{10}^{(5)\ast}(x)&=1049624576x^{10}-5248122880x^9+11386552320x^8-14057472000x^7+10881024000x^6-5484036096x^5+1808970240x^4-382018560x^3+48837600x^2-3359200x+92378.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-5.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(6)\ast}(x)&=1,\\
+    C_{1}^{(6)\ast}(x)&=24x-12,\\
+    C_{2}^{(6)\ast}(x)&=336x^2-336x+78,\\
+    C_{3}^{(6)\ast}(x)&=3584x^3-5376x^2+2520x-364,\\
+    C_{4}^{(6)\ast}(x)&=32256x^4-64512x^3+45696x^2-13440x+1365,\\
+    C_{5}^{(6)\ast}(x)&=258048x^5-645120x^4+612864x^3-274176x^2+57120x-4368,\\
+    C_{6}^{(6)\ast}(x)&=1892352x^6-5677056x^5+6773760x^4-4085760x^3+1302336x^2-205632x+12376,\\
+    C_{7}^{(6)\ast}(x)&=12976128x^7-45416448x^6+65286144x^5-49674240x^4+21450240x^3-5209344x^2+651168x-31824,\\
+    C_{8}^{(6)\ast}(x)&=84344832x^8-337379328x^7+567705600x^6-522289152x^5+285626880x^4-94381056x^3+18232704x^2-1860480x+75582,\\
+    C_{9}^{(6)\ast}(x)&=524812288x^9-2361655296x^8+4554620928x^7-4920115200x^6+3264307200x^5-1371009024x^4+361794048x^3-57302784x^2+4883760x-167960,\\
+    C_{10}^{(6)\ast}(x)&=3148873728x^{10}-15744368640x^9+34244001792x^8-42509795328x^7+33210777600x^6-16974397440x^5+5712537600x^4-1240436736x^3+164745504x^2-11938080x+352716.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-6.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(7)\ast}(x)&=1,\\
+    C_{1}^{(7)\ast}(x)&=28x-14,\\
+    C_{2}^{(7)\ast}(x)&=448x^2-448x+105,\\
+    C_{3}^{(7)\ast}(x)&=5376x^3-8064x^2+3808x-560,\\
+    C_{4}^{(7)\ast}(x)&=53760x^4-107520x^3+76608x^2-22848x+2380,\\
+    C_{5}^{(7)\ast}(x)&=473088x^5-1182720x^4+1128960x^3-510720x^2+108528x-8568,\\
+    C_{6}^{(7)\ast}(x)&=3784704x^6-11354112x^5+13601280x^4-8279040x^3+2681280x^2-434112x+27132,\\
+    C_{7}^{(7)\ast}(x)&=28114944x^7-98402304x^6+141926400x^5-108810240x^4+47604480x^3-11797632x^2+1519392x-77520,\\
+    C_{8}^{(7)\ast}(x)&=196804608x^8-787218432x^7+1328431104x^6-1230028800x^5+680064000x^4-228501504x^3+45224256x^2-4775232x+203490,\\
+    C_{9}^{(7)\ast}(x)&=1312030720x^9-5904138240x^8+11414667264x^7-12398690304x^6+8302694400x^5-3536332800x^4+952089600x^3-155054592x^2+13728792x-497420,\\
+    C_{10}^{(7)\ast}(x)&=8396996608x^{10}-41984983040x^9+91514142720x^8-114146672640x^7+89890504704x^6-46495088640x^5+15913497600x^4-3536332800x^3+484545600x^2-36610112x+1144066.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-7.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(8)\ast}(x)&=1,\\
+    C_{1}^{(8)\ast}(x)&=32x-16,\\
+    C_{2}^{(8)\ast}(x)&=576x^2-576x+136,\\
+    C_{3}^{(8)\ast}(x)&=7680x^3-11520x^2+5472x-816,\\
+    C_{4}^{(8)\ast}(x)&=84480x^4-168960x^3+120960x^2-36480x+3876,\\
+    C_{5}^{(8)\ast}(x)&=811008x^5-2027520x^4+1943040x^3-887040x^2+191520x-15504,\\
+    C_{6}^{(8)\ast}(x)&=7028736x^6-21086208x^5+25344000x^4-15544320x^3+5100480x^2-842688x+54264,\\
+    C_{7}^{(8)\ast}(x)&=56229888x^7-196804608x^6+284663808x^5-219648000x^4+97152000x^3-24482304x^2+3230304x-170544,\\
+    C_{8}^{(8)\ast}(x)&=421724160x^8-1686896640x^7+2853666816x^6-2656862208x^5+1482624000x^4-505190400x^3+102009600x^2-11075328x+490314,\\
+    C_{9}^{(8)\ast}(x)&=2998927360x^9-13495173120x^8+26146897920x^7-28536668160x^6+19262251008x^5-8302694400x^4+2273356800x^3-378892800x^2+34610400x-1307504,\\
+    C_{10}^{(8)\ast}(x)&=20392706048x^{10}-101963530240x^9+222670356480x^8-278900244480x^7+221159178240x^6-115573506048x^5+40129689600x^4-9093427200x^3+1278763200x^2-99985600x+3268760.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-8.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(9)\ast}(x)&=1,\\
+    C_{1}^{(9)\ast}(x)&=36x-18,\\
+    C_{2}^{(9)\ast}(x)&=720x^2-720x+171,\\
+    C_{3}^{(9)\ast}(x)&=10560x^3-15840x^2+7560x-1140,\\
+    C_{4}^{(9)\ast}(x)&=126720x^4-253440x^3+182160x^2-55440x+5985,\\
+    C_{5}^{(9)\ast}(x)&=1317888x^5-3294720x^4+3168000x^3-1457280x^2+318780x-26334,\\
+    C_{6}^{(9)\ast}(x)&=12300288x^6-36900864x^5+44478720x^4-27456000x^3+9108000x^2-1530144x+100947,\\
+    C_{7}^{(9)\ast}(x)&=105431040x^7-369008640x^6+535062528x^5-415134720x^4+185328000x^3-47361600x^2+6375600x-346104,\\
+    C_{8}^{(9)\ast}(x)&=843448320x^8-3373793280x^7+5719633920x^6-5350625280x^5+3009726720x^4-1037836800x^3+213127200x^2-23680800x+1081575,\\
+    C_{9}^{(9)\ast}(x)&=6372720640x^9-28677242880x^8+55667589120x^7-61009428480x^6+41467345920x^5-18058360320x^4+5016211200x^3-852508800x^2+79922700x-3124550,\\
+    C_{10}^{(9)\ast}(x)&=45883588608x^{10}-229417943040x^9+501851750400x^8-630899343360x^7+503327784960x^6-265391013888x^5+93301528320x^4-21498048000x^3+3090344400x^2-248648400x+8436285.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-9.svg"/>
+</p>
+
+$$\begin{align*}
+	C_{0}^{(10)\ast}(x)&=1,\\
+    C_{1}^{(10)\ast}(x)&=40x-20,\\
+    C_{2}^{(10)\ast}(x)&=880x^2-880x+210,\\
+    C_{3}^{(10)\ast}(x)&=14080x^3-21120x^2+10120x-1540,\\
+    C_{4}^{(10)\ast}(x)&=183040x^4-366080x^3+264000x^2-80960x+8855,\\
+    C_{5}^{(10)\ast}(x)&=2050048x^5-5125120x^4+4942080x^3-2288000x^2+506000x-42504,\\
+    C_{6}^{(10)\ast}(x)&=20500480x^6-61501440x^5+74314240x^4-46126080x^3+15444000x^2-2631200x+177100,\\
+    C_{7}^{(10)\ast}(x)&=187432960x^7-656015360x^6+953272320x^5-743142400x^4+334414080x^3-86486400x^2+11840400x-657800,\\
+    C_{8}^{(10)\ast}(x)&=1593180160x^8-6372720640x^7+10824253440x^6-10168238080x^5+5759353600x^4-2006484480x^3+418017600x^2-47361600x+2220075,\\
+    C_{9}^{(10)\ast}(x)&=12745441280x^9-57354485760x^8+111522611200x^7-122674872320x^6+83887964160x^5-36859863040x^4+10366836480x^3-1791504000x^2+171685800x-6906900,\\
+    C_{10}^{(10)\ast}(x)&=96865353728x^{10}-484326768640x^9+1061057986560x^8-1338271334400x^7+1073405132800x^6-570438156288x^5+202729246720x^4-47391252480x^3+6942078000x^2-572286000x+20030010.
+\end{align*}$$
+
+<p align="center">
+  <img src="images_svg/Shifted-Gegenbauer-Polynomials-10.svg"/>
+</p>
+
 ### Shifted Legendre polynomials
 
 $$P_n^\ast(x)=P_n(2x-1)$$