Home » Blog » Prove properties of the Bernoulli polynomials

# Prove properties of the Bernoulli polynomials

The Bernoulli polynomials are defined by

1. Find explicit formulas for the first Bernoulli polynomials in the cases .
2. Use mathematical induction to prove that is a degree polynomial in , where the degree term is .
3. For prove that .
4. For prove that

5. Prove that

for .

6. Prove that for ,

7. Prove that for ,

1. We start with the initial condition . This gives us

Now, using the integral condition to find ,

Thus,

Next, using this expression for we have

Using the integral condition to find ,

Thus,

Next, using this expression for we have

Using the integral condition to find ,

Thus,

Next, using this expression for we have

Using the integral condition to find ,

Thus,

Finally, using this expression for we have

Using the integral condition to find ,

Thus,

2. Proof. We have shown in part (a) that this statement is true for . Assume then that the statement is true for some positive integer , i.e.,

Then, by the definition of the Bernoulli polynomials we have,

where for . Then, taking the integral of this expression

Hence, the statement is true for the case ; hence, for all positive integers

3. Proof. From the integral property in the definition of the Bernoulli polynomials we know for ,

Then, using the first part of the definition we have ; therefore,

Thus, we indeed have

4. Proof. The proof is by induction. For the case we have

Therefore,

Since , the stated difference equation holds for . Assume then that the statement holds for some positive integer . Then by the fundamental theorem of calculus, we have

Therefore,

Hence, the statement is true for the case , and so it is true for all positive integers

5. Proof. (Let’s assume Apostol means for to be some positive integer.) First, we use the definition of the Bernoulli polynomials to compute the integral,

Now, we want to express the numerator as a telescoping sum and use part (d),

Thus, we indeed have

6. Proof.

Incomplete. I’ll try to fix parts (f) and (g) soon(ish).

1. Luca says:

I tried induction for (f):
For n=1 you have

*** QuickLaTeX cannot compile formula:
$P_1(1-x)=(1-x)-\frac{1}{2}=-x+\frac{1}{2}=-(x-\frac{1}{2})$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

and so the equality holds. Now suppose it holds for

*** QuickLaTeX cannot compile formula:
n=m\ge1

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

, and we want to show that

*** QuickLaTeX cannot compile formula:
$P_{m+1}(1-x)=(-1)^{m+1}P_{m+1}(x)$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

We observe that

*** QuickLaTeX cannot compile formula:
$\frac{d}{dx}\bigg(P_{m+1}(1-x)\bigg)=-P_{m+1}'(1-x)=-(m+1)P_m(1-x)$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

and that

*** QuickLaTeX cannot compile formula:
$\frac{d}{dx}\bigg((-1)^{m+1}P_{m+1}(x)\bigg)=(-1)^{m+1}(m+1)P_m(x)$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

and so the induction hypothesis gives us that

*** QuickLaTeX cannot compile formula:
$\frac{d}{dx}\bigg((-1)^{m+1}P_{m+1}(x)\bigg)=\frac{d}{dx}\bigg(P_{m+1}(1-x)\bigg)$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

Thus,

*** QuickLaTeX cannot compile formula:
P_{m+1}(1-x)

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

and

*** QuickLaTeX cannot compile formula:
(-1)^{m+1}P_{m+1}(x)

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

are primitives of the same function, and differ only by a constant. So to prove that they are equal, it suffices to exhibit one point of equality. \\
\\
If is even, then we have

*** QuickLaTeX cannot compile formula:
P_{m+1}(1-0)=P_{m+1}(0)

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

by part (c), so we’re done. \\
\\
If is odd, we wish to find a point at which

*** QuickLaTeX cannot compile formula:
$P_{m+1}(1-x)=-P_{m+1}(x) \iff P_{m+1}(1-x)+P_{m+1}(x)=0$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

By the definition, and making a substitution, we have

*** QuickLaTeX cannot compile formula:
$\int_{0}^{1}\bigg(P_{m+1}(1-x)+P_{m+1}(x)\bigg)dx=0$

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

Now I’ll refer back to an earlier exercise, exercise 7 from section 3.20, which tells us that if is non-negative and has zero integral over some interval, then it must be zero at every point on the interval where it’s continuous (the same holds if is non-positive). Since

*** QuickLaTeX cannot compile formula:
P_{m+1}

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

is continuous everywhere, we may apply the result here. There are two cases:
\\
)

*** QuickLaTeX cannot compile formula:
P_{m+1}(1-x)+P_{m+1}(x)

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

does not change sign on . Then by the above result, it must be zero everywhere, and we may exhibit any point on to complete the proof.
\\

*** QuickLaTeX cannot compile formula:
ii

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

)

*** QuickLaTeX cannot compile formula:
P_{m+1}(1-x)+P_{m+1}(x)

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

changes sign on . Then by the intermediate value theorem, it vanishes at some

*** QuickLaTeX cannot compile formula:
c\in[0,1]

*** Error message:
Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
These links might help in finding solution:
http://wordpress.org/extend/plugins/core-control/
http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

. In either case, we have found a point of equality.

• Luca says:

oh no something went wrong, here’s the latex:
I tried induction for (f):
For n=1 you have
$P_1(1-x)=(1-x)-\frac{1}{2}=-x+\frac{1}{2}=-(x-\frac{1}{2})$
and so the equality holds. Now suppose it holds for $n=m\ge1$, and we want to show that
$P_{m+1}(1-x)=(-1)^{m+1}P_{m+1}(x)$
We observe that
$\frac{d}{dx}\bigg(P_{m+1}(1-x)\bigg)=-P_{m+1}'(1-x)=-(m+1)P_m(1-x)$
and that
$\frac{d}{dx}\bigg((-1)^{m+1}P_{m+1}(x)\bigg)=(-1)^{m+1}(m+1)P_m(x)$
and so the induction hypothesis gives us that
$\frac{d}{dx}((-1)^{m+1}P_{m+1}(x))=\frac{d}{dx}(P_{m+1}(1-x))$
Thus, $P_{m+1}(1-x)$ and $(-1)^{m+1}P_{m+1}(x)$ are primitives of the same function, and differ only by a constant. So to prove that they are equal, it suffices to exhibit one point of equality. \\
\\
If $m+1$ is even, then we have $P_{m+1}(1-0)=P_{m+1}(0)$ by part (c), so we’re done. \\
\\
If $m+1$ is odd, we wish to find a point $x$ at which
$P_{m+1}(1-x)=-P_{m+1}(x) \iff P_{m+1}(1-x)+P_{m+1}(x)=0$
By the definition, and making a substitution, we have
$\int_{0}^{1}\bigg(P_{m+1}(1-x)+P_{m+1}(x)\bigg)dx=0$
Now I’ll refer back to an earlier exercise, exercise 7 from section 3.20, which tells us that if $f$ is non-negative and has zero integral over some interval, then it must be zero at every point on the interval where it’s continuous (the same holds if $f$ is non-positive). Since $P_{m+1}$ is continuous everywhere, we may apply the result here. There are two cases:
\\
$i$) $P_{m+1}(1-x)+P_{m+1}(x)$ does not change sign on $[0,1]$. Then by the above result, it must be zero everywhere, and we may exhibit any point on $[0,1]$ to complete the proof.
\\
$ii$) $P_{m+1}(1-x)+P_{m+1}(x)$ changes sign on $[0,1]$. Then by the intermediate value theorem, it vanishes at some $c\in[0,1]$. In either case, we have found a point of equality.

2. Mihajlo says:

f) can be proved by induction as well (e.g. similar to d) part).

• Mihajlo says:

Actually, while it is probably doable to do f) by induction (e.g. to prove that later odd Bernoulli numbers are 0), the other suggested solution here looks more appropriate.

• Anonymous says:

The proof of part (d) is wrong since he assumes that p_m+1(0)=0 which is clearly not true (in general) from part a