Is there a simple formula for this polynomial sum?

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The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about it here, and I would welcome any other related comments!

Let $Nequiv 3pmod4$ be a positive integer and let $M:=(N-1)/2$. Consider the following homogeneous polynomial of degree $N-1$:

$$T_N:=sum_r=0^M-1frac2 i^r X^r Y^N-1-rr!(N-1-r)!+fraci^M X^M Y^M(M!)^2inmathbbQ(i)[X,Y].$$

Question: Is there a simpler expression for $T_N$ that does not
involve summation notation, and depends only on $N$?

Of course, if the upper summation index in the definition of $T_N$ changed from $M-1$ to $N-1$ (call this modified version $T_N'$), we could use the binomial formula to write

$$T_N' = frac2(N-1)!(i X+Y)^N-1 + fraci^M X^M Y^M(M!)^2.$$

However in $T_N$ the summation index runs only up to $M-1$.

combinatorics polynomials summation binomial-coefficients

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edited Jul 16 at 16:12

asked Jul 16 at 15:08

Alex Saad

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up vote
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down vote

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The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about it here, and I would welcome any other related comments!

Let $Nequiv 3pmod4$ be a positive integer and let $M:=(N-1)/2$. Consider the following homogeneous polynomial of degree $N-1$:

$$T_N:=sum_r=0^M-1frac2 i^r X^r Y^N-1-rr!(N-1-r)!+fraci^M X^M Y^M(M!)^2inmathbbQ(i)[X,Y].$$

Question: Is there a simpler expression for $T_N$ that does not
involve summation notation, and depends only on $N$?

Of course, if the upper summation index in the definition of $T_N$ changed from $M-1$ to $N-1$ (call this modified version $T_N'$), we could use the binomial formula to write

$$T_N' = frac2(N-1)!(i X+Y)^N-1 + fraci^M X^M Y^M(M!)^2.$$

However in $T_N$ the summation index runs only up to $M-1$.

combinatorics polynomials summation binomial-coefficients

share|cite|improve this question

edited Jul 16 at 16:12

asked Jul 16 at 15:08

Alex Saad

1,434621

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about it here, and I would welcome any other related comments!

Let $Nequiv 3pmod4$ be a positive integer and let $M:=(N-1)/2$. Consider the following homogeneous polynomial of degree $N-1$:

$$T_N:=sum_r=0^M-1frac2 i^r X^r Y^N-1-rr!(N-1-r)!+fraci^M X^M Y^M(M!)^2inmathbbQ(i)[X,Y].$$

Question: Is there a simpler expression for $T_N$ that does not
involve summation notation, and depends only on $N$?

Of course, if the upper summation index in the definition of $T_N$ changed from $M-1$ to $N-1$ (call this modified version $T_N'$), we could use the binomial formula to write

$$T_N' = frac2(N-1)!(i X+Y)^N-1 + fraci^M X^M Y^M(M!)^2.$$

However in $T_N$ the summation index runs only up to $M-1$.

combinatorics polynomials summation binomial-coefficients

share|cite|improve this question

edited Jul 16 at 16:12

asked Jul 16 at 15:08

Alex Saad

1,434621

The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about it here, and I would welcome any other related comments!

Let $Nequiv 3pmod4$ be a positive integer and let $M:=(N-1)/2$. Consider the following homogeneous polynomial of degree $N-1$:

$$T_N:=sum_r=0^M-1frac2 i^r X^r Y^N-1-rr!(N-1-r)!+fraci^M X^M Y^M(M!)^2inmathbbQ(i)[X,Y].$$

Question: Is there a simpler expression for $T_N$ that does not
involve summation notation, and depends only on $N$?

Of course, if the upper summation index in the definition of $T_N$ changed from $M-1$ to $N-1$ (call this modified version $T_N'$), we could use the binomial formula to write

$$T_N' = frac2(N-1)!(i X+Y)^N-1 + fraci^M X^M Y^M(M!)^2.$$

However in $T_N$ the summation index runs only up to $M-1$.

combinatorics polynomials summation binomial-coefficients

share|cite|improve this question

edited Jul 16 at 16:12

asked Jul 16 at 15:08

Alex Saad

1,434621

share|cite|improve this question

share|cite|improve this question

edited Jul 16 at 16:12

edited Jul 16 at 16:12

edited Jul 16 at 16:12

asked Jul 16 at 15:08

Alex Saad

1,434621

asked Jul 16 at 15:08

Alex Saad

1,434621

asked Jul 16 at 15:08

Alex Saad

1,434621

1,434621

add a comment | 

add a comment | 

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