The Bernoulli polynomials are defined by
- Find explicit formulas for the first Bernoulli polynomials in the cases .
- Use mathematical induction to prove that is a degree polynomial in , where the degree term is .
- For prove that .
- For prove that
- Prove that
for .
- Prove that for ,
- Prove that for ,
- 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,
- 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
- 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
- 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
- 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
- Proof.
Incomplete. I’ll try to fix parts (f) and (g) soon(ish).
For part (f), I found a way to do it online. First, prove that if some polynomials {Qn} satisfy the three conditions above then the polynomial Qn(x) must equal Pn(x) for every n. Then define polynomial Qn(x) := (-1)^n Pn(1-x) and show that this polynomial satisfies the three conditions above. Part (g) follows from part (f) and part (c).
Hello, I think you have to add a constant after the indefinite integrals in (d). Thanks for your solutions!
Solution of letter f is in this link, page 8: https://www.mathi.uni-heidelberg.de/~theiders/PS-Analysis/Bernoullische_Polynome.pdf
With letter f proved, it’s trivial solve the part g.