Function and Characterization of poly(T) and (AT)
Poly(T) is an oligonucleotide with only repeating thymine residues. These are generally used as primers in Reverse Transcription PCR, as mature eukaryotic mRNAs have a Poly(A) [adenosine] tail added to them as a post transcriptional modification in order to protect against mRNA degradation. You would also sometimes see this referred to as an oligo dT. It could also be the primer for some paired end adapters used in next-generation sequencing (NGS)

The 45 generally would indicate the number of residues in the oligonucleotide.

Poly(A) could be a primer for a paired end adapter for NGS, however I would need to see the context of where you saw the terms to be sure.

d = Deoxyribonucleotide, in other words you are dealing with a DNA oligonucleotide (though Thymine is also an indication). Technically the d refers to the 2' Carbon on ribose having a hydrogen instead of a hydroxyl attached to it making it deoxyribose.

Without the d it is a Ribonucleotide, indicating that you are looking at an RNA molecule and that the 2' Carbon on ribose has a hydroxyl group.

You will sometimes also see ddN (N for any base). This indicates a chain terminating dideoxynucleotide which has a hydrogen on both the 2' and 3' Carbon instead of hydroxyl groups. The reason this molecule is chain terminating is because you need the 3' hydroxyl in order to link to a 5' phosphate group to form a phosphodiester bond, which is needed to continue the elongation of the nucleic acid molecule.

As for the ss(AT)15, I think you are correct in saying that this is a 15 pair repeat of AT so a 30 nucleotide oligonucleotide. Though it didn't say dAdT, you are likely correct that they are deoxyribonucleotides as they use thymine instead of uracil as the base.

Just in case, the ss stands for single-stranded, which is often the case for oligonucleotide primers for PCR reactions.