Math Is Fun Forum

pinkualvares99

Member

Registered: 2015-03-16

Posts: 1

help on generating function

Find the generating function for :
(1,5k(k+1)/2,25k(k+1)(k+2)/6,125k(k+1)(k+2)(k+3)/24,....)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

Hi;

Looks like that first term should be k giving the sum:

but the gf is still not possible in terms of elementary functions.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

If the first term is k it looks like expansion of

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

Hi;

Have you expanded that?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

Hi bobbym,
My old age is showing its sign. I took it for granted that it was k(k-1)/2 , etc. Interestingly, yesterday after my walk, I placed my umbrella on the chair and myself stood in the corner.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

Do not worry about such things. If a person is not making mistakes he is asleep.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

Hi bobbym,
The following must work. Unlike previous one it is infinite series.
If the first term is k it looks like expansion of

But there is no clarification from pinkualvares 99 whether the first term is 1 or k. we can fix that also by

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

How can that generate the sequence (k,5k(k+1)/2,25k(k+1)(k+2)/6,125k(k+1)(k+2)(k+3)/24,....)? Therefore I do not agree that that is the generating function.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

Last edited by thickhead (2016-08-03 05:26:59)

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

How do you get that out of (-4)^(-k)? The binomial theorem maybe. I do not think that is the generating function though.

Definition: A generating function when expanded is a power series in x that has the given sequence as coefficients.

Your use of the binomial theorem here has some problems for instance:

let us check your answer, you state

This is incorrect, supposing we plug in k = 2, on the LHS we get

now on the RHS just the first two terms alone give us

it should be obvious that the extra terms on the RHS are all positive so the RHS > 11. 

I think we can find many more values for k where A) will not hold too.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

Yes. I ignored the small print in binomial theorem (just as smokers ignore statutory warning on their cigarette pack) that |x| must be less than 1 for negative or fractional power.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

When dealing with analysis that is true but with generating functions this does apply but only when we need it to.

The coefficients of that gf are what we are in interested in. The series is only considered like a clothesline that the coefficients are hanging from. Questions of convergence do not apply.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: help on generating function

But  if we use some value like x=2 L.H.S=-1 but R.H.S. is definitely positive.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help on generating function

Yes, the answer you gave in #9 has that problem.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.