OEIS/Square Root Recurrences - Revision history

imported>Gfis: Mathar from arxiv.org

2022-05-14T09:58:07Z

Mathar from arxiv.org

← Older revision		Revision as of 09:58, 14 May 2022
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	Richard J.. Mathar wrote a '''[~~http~~://~~www~~.~~mpia.de/~mathar~~/~~public~~/~~fischer20200119~~.~~pdf~~ detailled paper]''' on this topic.		Richard J.. Mathar wrote a '''[https://arxiv.org/abs/2109.02112 detailled paper]''' on this topic.
	===Examples===		===Examples===
	====1 / Sqrt(1 - 2bx + d*x^2)====		====1 / Sqrt(1 - 2bx + d*x^2)====

imported>Gfis: MMA example A002426

2020-08-28T17:29:09Z

MMA example A002426

← Older revision		Revision as of 17:29, 28 August 2020
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	make runholo OFFSET=0 MATRIX="[[0],[36,-36],[2,-4],[0,1]]" INIT="[1,2]"		make runholo OFFSET=0 MATRIX="[[0],[36,-36],[2,-4],[0,1]]" INIT="[1,2]"
	new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);		new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);
			===[https://oeis.org/A002426 A002426] Central trinomial coefficients ===
			Get["SpecialFunctions.m"] (* from Wolfram Koepf, Kassel *)
			gf=1/(1 - 2x - 3x^2)^(1/2);
			de=HolonomicDE[gf,x]
			re=DEtoRE[de,f[x],a[k]]
			RecurrenceTable[{re,a[0]==1,a[1]==1},a,{k,0,20}]
			(* Out[45]= (1 + 3 x) f[x] + (1 + x) (-1 + 3 x) f'[x] == 0
			Out[46]= 3 (1 + k) a[k] + (3 + 2 k) a[1 + k] - (2 + k) a[2 + k] == 0
			{1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, ... *)
	===Order 1===		===Order 1===
	f = [1/Sqrt[1-b*x],x]] a(0) = 1 / 0!		f = [1/Sqrt[1-b*x],x]] a(0) = 1 / 0!

imported>Gfis at 20:58, 16 February 2020

2020-02-16T20:58:53Z

← Older revision		Revision as of 20:58, 16 February 2020
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			Richard J.. Mathar wrote a '''[http://www.mpia.de/~mathar/public/fischer20200119.pdf detailled paper]''' on this topic.
	===Examples===		===Examples===
	====1 / Sqrt(1 - 2bx + d*x^2)====		====1 / Sqrt(1 - 2bx + d*x^2)====
	Cf. [https://cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.pdf Noe], equation (4):		Cf. [https://cs.uwaterloo.a/journals/JIS/VOL9/Noe/noe35.pdf Noe], equation (4):
	[https://oeis.org/A098455 A098455]: b=2; d=-36;		[https://oeis.org/A098455 A098455]: b=2; d=-36;
	RecurrenceTable[{a[0]==1, a[1]==b, na[n]==(2n-1)ba[n-1] - (n-1)da[n-2]}, a[n], {n,0,8}]		RecurrenceTable[{a[0]==1, a[1]==b, na[n]==(2n-1)ba[n-1] - (n-1)da[n-2]}, a[n], {n,0,8}]

imported>Gfis: in general

2020-01-03T08:39:57Z

in general

← Older revision		Revision as of 08:39, 3 January 2020
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	new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);		new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);
	===Order 1===		===Order 1===
	f = [1/Sqrt[1-b*x],x]] a(0) = 1 / 0!		f = [1/Sqrt[1-b*x],x]] a(0) = 1 / 0!
	df/dx = b/(2(1 - bx)^(3/2)) a(1) = b/2 / 1!		df/dx = b/(2(1 - bx)^(3/2)) a(1) = b/2 / 1!
	(3b^2)/(4(1 - bx)^(5/2)) a(2) = 3/4b / 2!		(3b^2)/(4(1 - bx)^(5/2)) a(2) = 3/4b^2 / 2!
	(15b^3)/(8(1 - bx)^(7/2)) a(3) = 35/8*b / 3!		(15b^3)/(8(1 - bx)^(7/2)) a(3) = 35/8*b^3 / 3!
	(105b^4)/(16(1 - bx)^(9/2)) a(4) = 357/2^4b / 4!		(105b^4)/(16(1 - bx)^(9/2)) a(4) = 357/16b^4 / 4!
	=> '''2na(n) - b(2n-1)*a(n-1) = 0'''		=> '''2na(n) - (2n-1)b*a(n-1) = 0'''
	===Order 2===		===Order 2===
	1/(1-bx-cx^2)^(1/2)		1/(1-bx-cx^2)^(1/2)
	~~=> 0! * a(0) = 1~~
	-(-b-2cx)/(2(1-bx-c*x^2)^(3/2))		-(-b-2cx)/(2(1-bx-c*x^2)^(3/2))
	~~=> 1! * a(1) = b/2~~
	(3(-b-2cx)^2)/(4(1-bx-cx^2)^(5/2))		(3(-b-2cx)^2)/(4(1-bx-cx^2)^(5/2))
	+ c/(1-bx-cx^2)^(3/2)		+ c/(1-bx-cx^2)^(3/2)
	~~=> 2! * a(2) = 3*b^2/4 + c~~
	(-15(-b-2cx)^3)/(8(1-bx-cx^2)^(7/2))		(-15(-b-2cx)^3)/(8(1-bx-cx^2)^(7/2))
	- (9c(-b-2cx))/(2(1-bx-c*x^2)^(5/2))		- (9c(-b-2cx))/(2(1-bx-c*x^2)^(5/2))
	~~=> 3! * a(3) = 15/8b^3 + 9b*c/2~~
	(105(-b-2cx)^4)/(16(1-bx-cx^2)^(9/2))		(105(-b-2cx)^4)/(16(1-bx-cx^2)^(9/2))
	+ (45c(-b-2cx)^2)/(2(1-bx-c*x^2)^(7/2))		+ (45c(-b-2cx)^2)/(2(1-bx-c*x^2)^(7/2))
	+ (9c^2)/(1-bx-c*x^2)^(5/2)		+ (9c^2)/(1-bx-c*x^2)^(5/2)
	=> 4! * a(4) = ~~105~~/16b^4 + 45/2b^2c^2 + 9c^2
			0! * a(0) = 1
			1! * a(1) = 1/2*b
			2! * a(2) = 13/4b^2 + c = 3/2ba(1) + c
			3! * a(3) = 135/8b^3 + 9/2b*c
			4! * a(4) = 1357/16b^4 + 45/2b^2c^2 + 9*c^2
			=> '''1(2n-0)a(n) - (2n-1)ba(n-1) - (2n-2)c*a(n-2) = 0'''

			a(1) = 1/2*b
			a(2) = 3/2ba(1) + 2/2*c
			a(3) = 5/2ba(2) + 4/2bc + 2/2*d

			In general for g.f. =
			(1 - c1x - c2x^2 ... - ck*x^k )^(-1/2)
			1(2n-0)a(n) - (2n-1)c1a(n-1) - (2n-2)c2a(n-2) ... - (2n-k)cka(n-k) = 0

imported>Gfis: Order 2

2020-01-01T11:33:06Z

Order 2

New page

===Examples===
====1 / Sqrt(1 - 2*b*x + d*x^2)====
Cf. [https://cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.pdf Noe], equation (4):
[https://oeis.org/A098455 A098455]: b=2; d=-36;
RecurrenceTable[{a[0]==1, a[1]==b, n*a[n]==(2*n-1)*b*a[n-1] - (n-1)*d*a[n-2]}, a[n], {n,0,8}]
-> 1, 2, 24, 128, 1096, 7632, 60864, 461568, 3648096
make runholo OFFSET=0 MATRIX="[[0],[36,-36],[2,-4],[0,1]]" INIT="[1,2]"
new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);
===Order 1===
f = [1/Sqrt[1-b*x],x]] a(0) = 1 / 0!
df/dx = b/(2*(1 - b*x)^(3/2)) a(1) = b/2 / 1!
(3*b^2)/(4*(1 - b*x)^(5/2)) a(2) = 3/4*b / 2!
(15*b^3)/(8*(1 - b*x)^(7/2)) a(3) = 3*5/8*b / 3!
(105*b^4)/(16*(1 - b*x)^(9/2)) a(4) = 3*5*7/2^4*b / 4!
=> '''2*n*a(n) - b*(2*n-1)*a(n-1) = 0'''
===Order 2===
1/(1-b*x-c*x^2)^(1/2)
=> 0! * a(0) = 1
-(-b-2*c*x)/(2*(1-b*x-c*x^2)^(3/2))
=> 1! * a(1) = b/2
(3*(-b-2*c*x)^2)/(4*(1-b*x-c*x^2)^(5/2))
+ c/(1-b*x-c*x^2)^(3/2)
=> 2! * a(2) = 3*b^2/4 + c
(-15*(-b-2*c*x)^3)/(8*(1-b*x-c*x^2)^(7/2))
- (9*c*(-b-2*c*x))/(2*(1-b*x-c*x^2)^(5/2))
=> 3! * a(3) = 15/8*b^3 + 9*b*c/2
(105*(-b-2*c*x)^4)/(16*(1-b*x-c*x^2)^(9/2))
+ (45*c*(-b-2*c*x)^2)/(2*(1-b*x-c*x^2)^(7/2))
+ (9*c^2)/(1-b*x-c*x^2)^(5/2)
=> 4! * a(4) = 105/16*b^4 + 45/2*b^2*c^2 + 9*c^2