Thanatipanonda’s shifted binomial power linearization

Abstract

Every positive power of choose(n+dk,k) has an integral expansion in choose(n+dj,j), with coefficients generated by one integer recursion independent of the power index.

Definition 1.1 (The shifted binomial family).

$$\forall d\in \mathbb{N},\forall j\in \mathbb{N},\forall n\in \mathbb{N},B\left(d,j,n\right)=(\mathrm{choose}\left(n+d\cdot j,j\right):\mathbb{Z})$$

Formalization. D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.B (✓ std3).

Citation. Thotsaporn Thanatipanonda (2014). Beyond Zudilin’s Conjectured q-analog of Schmidt’s problem. DOI: 10.1080/10236198.2014.917635. URL: https://arxiv.org/abs/1403.4962v1.

Commentary.

The argument and the shift parameters are natural numbers. The binomial coefficient is computed in Nat and then cast to Int; the power and the expansion are evaluated in Int.

Definition 1.2 (The coefficient recursion).

$$\forall k\in \mathbb{N},\forall T\in \mathbb{N}\to \left(\mathbb{N}\to \mathbb{Z}\right),\forall j\in \mathbb{N},(a\left(k,T,0,j\right)=0)\wedge ((a\left(k,T,1,j\right)=ifj=kthen1else0)\wedge (\forall r\in \mathbb{N},a\left(k,T,r+2,j\right)=\sum _{i\in \mathrm{range}\left((r+1)\cdot k+1\right)}(T\left(j,i\right)\cdot a\left(k,T,r+1,i\right))))$$

Formalization. D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.a (✓ std3).

Citation. Thotsaporn Thanatipanonda (2014). Beyond Zudilin’s Conjectured q-analog of Schmidt’s problem. DOI: 10.1080/10236198.2014.917635. URL: https://arxiv.org/abs/1403.4962v1.

Commentary.

The initial coefficients are the Kronecker delta at k. For positive powers the next row is obtained using T(j,i), which has no power index. The unused row at power zero is defined to be zero. All sums use Finset.range(m) = {0,…,m-1}.

Definition 1.3 (Conjecture 4.1 with finite support).

$$claim\iff (\forall d\in \mathbb{N},\forall k\in \mathbb{N},\exists T\in \mathbb{N}\to \left(\mathbb{N}\to \mathbb{Z}\right),(\forall i\in \mathbb{N},\forall j\in \mathbb{N},(i+k<j)\Rightarrow (T\left(j,i\right)=0))\wedge (\forall r\in \mathbb{N},(1\le r)\Rightarrow ((\forall j\in \mathbb{N},(r\cdot k<j)\Rightarrow (a\left(k,T,r,j\right)=0))\wedge (\forall n\in \mathbb{N},B{\left(d,k,n\right)}^r=\sum _{j\in \mathrm{range}\left(r\cdot k+1\right)}(a\left(k,T,r,j\right)\cdot B\left(d,j,n\right))))))$$

Formalization. D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.claim (✓ std3).

Citation. Thotsaporn Thanatipanonda (2014). Beyond Zudilin’s Conjectured q-analog of Schmidt’s problem. DOI: 10.1080/10236198.2014.917635. URL: https://arxiv.org/abs/1403.4962v1.

Commentary.

Conjecture 4.1 (arXiv:1403.4962v1, p. 6): “For any integers d, k ≥ 0 and r ≥ 1, there exist integers $a_{d,k,j}^{(r)}$ such that $(\begin{array}{c}n+dk\\ k\end{array}{)}^r={\sum }_j{a}_{d,k,j}^{(r)}(\begin{array}{c}n+dj\\ j\end{array})$ for all n = 0, 1, 2, …. Moreover $a_{d,k,j}^{(r)}$ can be defined as following: $a_{d,k,k}^{(1)}=1$, $a_{d,k,j}^{(1)}=0$ for j ≠ k and $a_{d,k,j}^{(r+1)}={\sum }_i{S}_d(k,j,i)a_{d,k,i}^{(r)}$, where, $S_d(k,j,i)$ are integers, independent of r, for all d, k, j, i.”

The encoding uses d,k,n,j,i,r in Nat and T(j,i) in Int for S_d(k,j,i), with d and k fixed before T is chosen. The coefficients are a(k,T,r,j). The support clauses justify replacing the source sums by ranges ending at rk and i+k. Both sentences are included; no positivity of the integer coefficients is asserted.

Theorem 1.4 (Integral linearization for every nonnegative shift).

$$\forall d\in \mathbb{N},\forall k\in \mathbb{N},\exists T\in \mathbb{N}\to \left(\mathbb{N}\to \mathbb{Z}\right),(\forall i\in \mathbb{N},\forall j\in \mathbb{N},(i+k<j)\Rightarrow (T\left(j,i\right)=0))\wedge (\forall r\in \mathbb{N},(1\le r)\Rightarrow ((\forall j\in \mathbb{N},(r\cdot k<j)\Rightarrow (a\left(k,T,r,j\right)=0))\wedge (\forall n\in \mathbb{N},B{\left(d,k,n\right)}^r=\sum _{j\in \mathrm{range}\left(r\cdot k+1\right)}(a\left(k,T,r,j\right)\cdot B\left(d,j,n\right)))))$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.result (✓ std3). ∎

Resolves. Problems/thanatipanonda-2014-shifted-binomial-power-linearization (proved) by D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.result.

Source. Repository-derived.

Acknowledgement. Thotsaporn Thanatipanonda (2014). Beyond Zudilin’s Conjectured q-analog of Schmidt’s problem. DOI: 10.1080/10236198.2014.917635. URL: https://arxiv.org/abs/1403.4962v1.

Commentary.

Vandermonde’s identity expresses each B(d,t,n) as an integer combination of choose(n,v), with diagonal coefficient one. Strong induction constructs an integral triangular inverse. The binomial product identity and this inverse give integer structure constants for B(d,k,n) times B(d,i,n), supported at j≤i+k. Induction on the positive power then gives the fixed recursion and the bound j≤rk. Conjecture 4.2 on holonomicity is a separate question.

References

• Truth anchor: D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.B
• Truth anchor: D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.a
• Truth anchor: D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.claim
• Truth anchor: D5/S3/Combinatorics/BinomialBases/ShiftedBinomialPowerLinearization.result