2. Laguerre Polynomials The Laguerre polynomials can be calculated via the Rodrigues formula d" re². n! dr" 1 L„ (x) (e*x"). %3D a) Use the general Leibniz rule d" dn-k dk k k=0 drk9 to proof that L„ (1) = E 'n(-x)* | %3D k k=0 k!
2. Laguerre Polynomials The Laguerre polynomials can be calculated via the Rodrigues formula d" re². n! dr" 1 L„ (x) (e*x"). %3D a) Use the general Leibniz rule d" dn-k dk k k=0 drk9 to proof that L„ (1) = E 'n(-x)* | %3D k k=0 k!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Please show all steps
![2. Laguerre Polynomials
The Laguerre polynomials can be calculated via the Rodrigues formula
dn
(e-*r") .
1
(x) "7
n! dæn
a) Use the general Leibniz rule
d"
S(z)9(2)] = )
dn-k
dk
dr"
drn-k (.
k=0
to proof that
(- x)*
k
Ln(z) %3D Σ
k!
k=0
b) Kummer's function, also known as confluent hypergeometric function, is defined via
q(n)
1F; (a; b; x) = D
b(n) n!
n=0
where we have used the rising factorial a") also known as Pochhammer function. Proof that
d
d F: (a; b; x) = 7 1F (a + 1; b+ 1; x).
a
%3D
c) Use Eqs. (5) and (6) to show that
Ln (x) = 1F;(– n; 1; x).
d) With the help of Eqs. (7) and (8) proof that
dm
n!
dæm Ln (x) = ( – 1)™,
(n
1Fi(- n+ m; m +1; x).
m)! m!
-
e) The generating function of the Laguerre polynomials is
E(1.t) := .
-e
1- t
(1
Show that
E(x,t) = L (x)t".
(1
n=0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0329f04-48c0-4c72-87ef-c458d4687c8a%2F160a8077-169a-4c81-887b-81edf199a2ba%2Fbt3e1n5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Laguerre Polynomials
The Laguerre polynomials can be calculated via the Rodrigues formula
dn
(e-*r") .
1
(x) "7
n! dæn
a) Use the general Leibniz rule
d"
S(z)9(2)] = )
dn-k
dk
dr"
drn-k (.
k=0
to proof that
(- x)*
k
Ln(z) %3D Σ
k!
k=0
b) Kummer's function, also known as confluent hypergeometric function, is defined via
q(n)
1F; (a; b; x) = D
b(n) n!
n=0
where we have used the rising factorial a") also known as Pochhammer function. Proof that
d
d F: (a; b; x) = 7 1F (a + 1; b+ 1; x).
a
%3D
c) Use Eqs. (5) and (6) to show that
Ln (x) = 1F;(– n; 1; x).
d) With the help of Eqs. (7) and (8) proof that
dm
n!
dæm Ln (x) = ( – 1)™,
(n
1Fi(- n+ m; m +1; x).
m)! m!
-
e) The generating function of the Laguerre polynomials is
E(1.t) := .
-e
1- t
(1
Show that
E(x,t) = L (x)t".
(1
n=0
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
