# Laguerre Polynomials

Laguerre polynomials are solutions to Laguerre's equation, a second-order linear differential equation: $$\xy''+(1-x)y'+ny=0\$$. For a given value of n, the solution, y, is named $$\L_n(x)\$$.

The polynomials can be found without calculus using recursion:

$$\L_0(x)=1\$$

$$\L_1(x)=1-x\$$

$$\L_{k+1}(x)=\frac{(2k+1-x)L_k(x)-kL_{k-1}(x)}{k+1}\$$

Summation can be used to the same end:

$$\L_n(x)=\sum\limits_{k=0}^{n}{n\choose k}\frac{(-1)^k}{k!}x^k\$$

$$\L_n(x)=\sum\limits_{i=0}^n\prod\limits_{k=1}^i\frac{-(n-k+1)x}{k^2}\$$

The first Laguerre polynomials are as follows:

Coefficients can be found here.

# The Challenge

Given a nonnegative integer n and a real number x, find $$\L_n(x)\$$.

# Rules

• This is so the shortest answer in bytes wins.

• Assume only valid input will be given.

• Error should be under one ten-thousandth (±0.0001) for the test cases.

# Test Cases

Here, n is the first number and x is the second.

In: 1 2
Out: -1

In: 3 1.416
Out: -0.71360922

In: 4 8.6
Out: −7.63726667

In: 6 -2.1
Out: 91.86123261

• I like that this challenge asks for the value of the polynomial on a certain input rather than its list of coefficients.
– xnor
Jul 13, 2020 at 8:17
• Thanks @xnor, it made more sense that way, being a polynomial and all Jul 13, 2020 at 8:19
• The point of writing "real" was to imply that the imaginary part of the input is always 0. Any real number can be approximated by a rational number by the limiting process because Q is a dense set. To change the word "real" to "rational" is unnecessary. @AdHocGarfHunter Jul 13, 2020 at 21:57
• Certainly @AdHocGarfHunter Jul 13, 2020 at 22:40
• Another useful fact about Laguerre polynomials: $$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}( e^{-x} x^n)$$ Jul 15, 2020 at 17:24

# Python 2, 53 bytes

f=lambda n,x:n<1or((2*n-1-x)*f(n-1,x)-~-n*f(n-2,x))/n


Try it online!

• Outputs True if n=0 Jul 13, 2020 at 8:37
• @fireflame241 Since True is basically the same as 1 in Python, this is allowed. See this meta post for details.
– ovs
Jul 13, 2020 at 9:57
• 52 bytes in Python 3.8.
– ovs
Jul 13, 2020 at 9:58

# Wolfram Language (Mathematica), 9 bytes

LaguerreL


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• 7 bytes: lagrrel Jul 13, 2020 at 8:33
• Wow, I did not expect a built-in to be golfable. Jul 13, 2020 at 8:35
• lagrrel works on WolframAlpha but not in Mathematica. Jul 14, 2020 at 7:12
• oh darn, I thought WA and mathematica were identical Jul 14, 2020 at 23:11

# Jelly, 11 bytes

cŻ÷Ż!$ƲṚḅN}  A dyadic Link accepting $$\n\$$ on the left and $$\x\$$ on the right which yields $$\L_n(x)\$$. Try it online! ### How? This makes the observation that $$\L_n(x)=\sum\limits_{k=0}^{n}{n\choose k}\frac{(-1)^k}{k!}x^k=\sum\limits_{k=0}^{n}{(-x)^k}\frac{n\choose k}{k!}\$$ which is the evaluation of a base $$\-x\$$ number with n+1 digits of the form $$\\frac{n\choose k}{k!}\$$. cŻ÷Ż!$ƲṚḅN} - Link: n, x
Ʋ     - last four links as a monad - f(n):
Ż          -   zero-range (n) -> [0, 1, 2, ..., n]
c           -   (n) binomial (that) -> [nC0, nC1, nC2, ..., nCn]
$- last two links as a monad - g(n): Ż - zero-range (n) -> [0, 1, 2, ..., n] ! - factorial (that) -> [0!, 1!, 2!, ..., n!] ÷ - division -> [nC0÷0!, nC1÷1!, nC2÷2!, ..., nCn÷n!] Ṛ - reverse -> [nCn÷n!, ..., nC2÷2!, nC1÷1!, nC0÷0!] } - use the chain's right argument for: N - negate -> -x ḅ - convert from base (-x) -> -xⁿnCn÷n!+...+-x²nC2÷2!+-x¹nC1÷1!+-x°nC0÷0!  # MATL, 5 bytes _1iZh  Inputs are $$\n\$$, then $$\x\$$. Try it online! Or verify all test cases. ### How it works This uses the equivalence of Laguerre polynomials and the (confluent) hypergeometric function: $$\ L_n(x) = {} _1F_1(-n,1,x) \$$ _ % Implicit input: n. Negate 1 % Push 1 i % Input: x Zh % Hypergeometric function. Implicit output  # JavaScript (ES6), 48 42 41 bytes Expects (x)(n). May output true instead of 1. x=>g=k=>k<1||((x-k---k)*g(k)+k*g(k-1))/~k  Try it online! # Python 3.8 (pre-release), 66 bytes L=lambda n,x:((2*n-1-x)*L(d:=n-1,x)-d*L(n-2,x))/n if n>1else 1-n*x  Try it online! Direct implementation of the recursive algorithm, with one interesting part: L(1,x) and L(0,x) can be combined as L(n,x)=1-n*x. Could save 2 bytes using L=lambda n,x:n>1and((2*n-1-x)*L(d:=n-1,x)-d*L(n-2,x))/n or 1-n*x, but L(n) is not necessarily zero. # APL (Dyalog Unicode), 16 bytes 1⊥⍨0,⎕×(-÷⌽×⌽)⍳⎕  Try it online! A full program that takes n and x from two separate lines of stdin. ### How it works 1⊥⍨0,⎕×(-÷⌽×⌽)⍳⎕ ⍳⎕ ⍝ Take n and generate 1..n (-÷⌽×⌽) ⍝ Compute i÷(n+1-i)^2 for i←1..n 0,⎕× ⍝ Multiply x to each and prepend 0, call it B 1⊥⍨ ⍝ Convert all ones from base B to single number  The mixed base conversion looks like this: 1..n: ... n-3 n-2 n-1 1 B: 0 ... (n-3)x/4^2 (n-2)x/3^2 (n-1)x/2^2 nx digits: 1 ... 1 1 1 1 digit values: x^n/n! ... (nC3 x^3/3!) (nC2 x^2/2!) (nC1 x^1/1!) (nC0 x^0/0!)  It is essentially a fancy way to write the sum of product scan over 1, nx, (n-1)x/2^2, (n-2)x/3^2, .... This happens to be shorter than a more straightforward -x-base conversion (evaluating a polynomial at -x): # APL (Dyalog Unicode), 18 bytes (-⎕)⊥⌽1,(!÷⍨⊢!≢)⍳⎕  Try it online! • Functional version: 1⊥⍨0,×∘(-÷⌽×⌽)∘⍳ – Adám Jul 13, 2020 at 14:36 # Python 3.8 (pre-release), 61 bytes L=lambda k,x:k<1or[1-x,L(w:=k-1,x)*(k+w-x)-L(k-2,x)*w][k>1]/k  Try it online! # JavaScript (Node.js), 36 bytes x=>(i=0,g=n=>n?1-x*n/++i/i*g(n-1):1)  Try it online! Just convert the formula to this, and use recursive: $$L_n(x) = \sum_{i=0}^n\prod_{k=1}^i\frac{-(n-k+1)x}{k^2}$$ # J, 37 20 bytes -5 thanks to @Bubbler Calculates the polynomial adapted from the summation formula and uses J's p. operator to calculate that polynomial with a given x. (p.-)~i.((!]/)%!)@,]  Try it online! # J, 45 byte Alternative Recursive function. 1:-@.[~ ::((>:@]%~($:*[-~1+2*])-]*(\$:<:))<:)


Try it online!

### How it works

We define a hook (fg), which is x f (g n). f is (p.-)~ so it will be evaluated as ((i.((!]/)%!)@,]) n) p. (- x).

(p.-)~i.((!]/)%!)@,]
i.         @,] enumerate 3 -> 0 1 2, append 3 -> 0 1 2 3, …
(!]/)       3 over i
%      divided by
!     !i
-                 negate x
p.                  apply -x to the polynomial expressed in J as
1 3 1.5 0.166667, so 1-3(-x)+1.5(-x)^2+0.16(-x)^3

• Top solution can be golfed to 20 bytes. Jul 13, 2020 at 23:50
• @Bubbler all great tips I never would have thought of. Thanks!
– xash
Jul 14, 2020 at 10:56

# Pari/GP, 39 bytes

Using the formula $$\L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k\$$.

l(n,x)=sum(k=0,n,n!*(-x)^k/(n-k)!/k!^2)


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# Pari/GP, 45 bytes

Using the generating function $$\\sum_{n=0}^\infty x^n L_n(t)= \frac{1}{1-x} e^{-xt/(1-x)}\$$.

l(n,t)=Vec(exp(-x*t/(1-x)+O(x^n++))/(1-x))[n]


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# Charcoal, 29 bytes

⊞υ¹ＦＮ⊞υ×⌈υＬυＩ↨Ｅυ∕⌈υ×ιＸ§⮌υκ²±Ｎ


Try it online! Link is to verbose version of code. Uses a slightly modified version of the summation given in the question. Explanation:

⊞υ¹ＦＮ⊞υ×⌈υＬυ


Calculate the factorials from $$\0!\$$ to $$\n!\$$.

Ｉ↨Ｅυ∕⌈υ×ιＸ§⮌υκ²±Ｎ


For each index $$\i\$$ from $$\0\$$ to $$\n\$$ calculate $$\\frac{n!}{i!(n-i)!^2}\$$ and then perform base conversion from base $$\-x\$$ which multiplies each term by $$\(-1)^{n-i}x^{n-i}\$$ and takes the sum.

If we set $$\k=n-i\$$ we see that we calculate $$\\sum\limits_{k=0}^{n}{\frac{n!(-1)^k}{(n-k)!k!^2}x^k}=\sum\limits_{k=0}^{n}{n\choose k}\frac{(-1)^k}{k!}x^k\$$ as required.

# 05AB1E, 16 bytes

1λèN·<I-₁*N<₂*-N/


Try it online. (No test suite for all test cases at once, since there seems to be a bug in the recursive environment..)

Explanation:

 λ                # Create a recursive environment
è               # to get the 0-based n'th value afterwards
# (where n is the first implicit input)
# (which will be output implicitly as result in the end)
1                 # Starting with a(-1)=0 and a(0)=1,
# and for every other a(N), we'll:
#  (implicitly push a(N-1))
N·             #  Push N doubled
<            #  Decrease it by 1
I-          #  Decrease it by the second input x
*         #  Multiply it by the implicit a(N-1)
N<       #  Push N-1
₂*     #  Multiply it by a(N-2)
-    #  Decrease the a(N-1)*(2N-1-x) by this (N-1)*a(N-2)
N/  #  And divide it by N: (a(N-1)*(2N-1-x)-(N-1)*a(N-2))/N


# Japt-x, 28 27 26 bytes

ò@l *VpX /Xl ²*JpX /(U-X l


Try it

# Japt, 30 29 28 bytes

ò x@l *VpX /Xl ²*JpX /(U-X l


Try it

## Explanation

ò x@l *VpX /Xl ²*JpX /(U-X l
ò                               // Create a array [0, 1, ..., U]
x                             // sum the array after mapping through
@                            // Function(X)
l                           //    U!
*VpX                      //    times V ** X
/Xl ²                //    divided by X! ** 2
*JpX            //    times (-1) ** X
/(U-X l    //    divided by (U - X)!

• U is the first input
• V is the second input
• ** represents exponentiation
• ! represents factorial

# C (gcc), 91 bytes

i;k;float f(n,x)float x;{float p,s=0;for(i=++n;k=i--;s+=p)for(p=1;--k;)p*=(k-n)*x/k/k;x=s;}


Try it online!

Straighforward implementation of polynomial expansion. Slightly golfed less

i;k;
float f(n,x)float x;{
float p,s=0;
for(i=++n;k=i--;s+=p)
for(p=1;--k;)
p*=(k-n)*x/k/k;
x=s;
}


## Fortran (GFortran), 69 68 bytes

read*,n,a
print*,sum([(product([((j-n-1)*a/j/j,j=1,i)]),i=0,n)])
end
`

-1 byte thanks to @ceilingcat

The program reads in an implicit integer n and real a. Summation and product operations are performed using arrays (initialized using implicit loops) with the intrinsics sum() and product().

Try it online!